Cooperative Problem Solving in Physics A User's Manual

Cooperative Problem Solving in PhysicsA User’s ManualWhy? What? How?STEP 1STEP 2STEP 3STEP 4STEP 5Recognize the ProblemWhat's going on?Describe the problem in terms of the fieldWhat does this have to do with ...... ?Plan a solutionHow do I get out of this?Execute the planLet's get an answerEvaluate the solutionCan this be true?Kenneth HellerPatricia HellerUniversity of MinnesotaWith support from the National Science Foundation, University of Minnesota, and U.S.Department of Education© Kenneth & Patricia Heller, 2010

AcknowledgmentsIn reaching this stage in this work, we gratefully acknowledge the support of the University ofMinnesota, the U.S. Department of Education FIPSE program, and the National ScienceFoundation. This work would not have existed without the close cooperation of the Universityof Minnesota School of Physics and Astronomy and Department of Curriculum and Instruction.We have incorporated the suggestions of many faculty members from both Physics andEducation at the University of Minnesota and other institutions that have communicated with usat workshops, meetings, and by e-mail. This work has depended on the efforts and feedback ofmany graduate student teaching assistants in the School of Physics and Astronomy over theyears. Much of this development is directly based on the research of the graduate students inthe University of Minnesota Physics Education Program: Jennifer Blue, Tom Foster, CharlesHenderson, Mark Hollabough, Ron Keith (deceased), and Laura McCullough. We are indebtedto our colleagues in the Physics Education community for their ideas, insights, and research.Finally we would like to thank Arnold Arons for his inspiration and guidance that has lasted alifetime.

Table of ContentsChapter 1. Introduction 1How to Use This Book 3Course Content and Structure for Cooperative Problem Solving 4Implementing Cooperative Problem Solving 5Our Laws of Instruction and other Frequently Used Icons 9Part 1 – Teaching Physics Through Problem Solving 13Chapter 2. Connecting Students, Physics, and Problem Solving 15Why Teach Physics through Problem Solving? 17What Students Typically Learn Through Problem Solving 18Who Benefits? 22Chapter 3. Combating Problem Solving that Avoids Physics 27How Context-rich Problems Help Students Engage in Real Problem Solving 28The Relationship Between Students’ Problem Solving Difficulties and the Designof Context-Rich Problems 31How to Discourage Students from Memorizing Formulas 33Chapter 4. Building Physics into Problem Solving 37What Is a Problem-solving Framework? 38A Physics Example: The Competent Problem-solving Framework 40Example of an “Ideal” Student’s Problem Solution 44Problem Solving as A Series of Translations 53Initial Objections to Teaching a Problem-solving Framework 55Chapter 5. Reinforcing Student Use of a Problem-solving Framework 59Flow-Charts of Problem-solving Decisions 61Instructor Problem Solutions on Answer Sheets 68Putting Together the Scaffolding for Students 68Chapter 6. Why Cooperative Problem Solving (CPS) 75The Dilemma and a Solution 75A Framework for Teaching and Learning Introductory Physics:Cognitive Apprenticeship 76Course Structures and Cognitive Apprenticeship 83Part 2 – Using Cooperative Problem Solving 87Chapter 7. Cooperative Problem Solving: Not Just Working in Groups 89Two Examples: Traditional Groups and Cooperative Groups 89Elements of Cooperative Learning 91Achievement for Traditional and Cooperative Groups: Summary 94Chapter 8. Group Problems and Managing Groups 97Group Structure and Management 97Appropriate Group Problems 105Chapter 9. Course Structure for Cooperative Problem-solving Sessions 109Scheduling: Continuity of Group Interactions 110What Resources Are Needed to Implement CPS 111i

Appropriate Course Grading 115Consistent Grading of Problem Solutions 117Getting Started 120Chapter 10. Results for Partial and Full Implementation of CPS 125Improvement in Problem Solving Performance 125Improvement in Knowledge Organization 130Improvement in Students’ Conceptual Understanding of Forces and Motionas measured by the Force Concept Inventory (FCI) 132Improvement in Students’ Conceptual Understanding of Forces and Motionas measured by Open Response Written Questions 135Part 3 – Teaching a CPS Session 145Chapter 11. Preparation for CPS Sessions 147Assign Students to Groups with Roles 149Prepare Problem & Information Sheet 150Write the Problem Solution 152Prepare an Answer Sheet (Optional) 154Prepare Group Functioning Evaluation Sheets (as Needed) 154Prepare Materials 157Chapter 12. Teaching a CPS Session 159Opening Moves (Steps 0 – 2) 159Middle Game (Steps 3 – 4) 161End Game (Steps 5 – 7) 163Chapter 13. Coaching Students During Group Work 165Monitor and Diagnose 166Intervene and Coach 166Coaching Dysfunctional Groups 167Coaching Groups with Physics Difficulties 171Part 4 – Personalizing a Problem-solving Framework, Problems, and Solutions 175Chapter 14. Personalize a Problem Solving Framework 177Step 1. Clarify the Situation 179Step 2. Gather Additional Information 183Step 3. Construct Your Problem-solving Framework 184Step 4. Reconcile Your Framework with Your Textbook Framework (if necessary) 189Step 5. Evaluate Your Framework 191Chapter 15. Building Context-Rich Problems from Textbook Problems 193Review of the Properties of Context-rich Problems 194Constructing a Context-rich Problem from a Textbook Problem 199Practice Building Context-rich Problems from Textbook Problems 204Chapter 16. Judging Context-Rich Problem Suitabilityfor Individual or Group Work 211Problem Difficulty Traits 212Decision Strategy for Judging the Suitability of Problems for Their Intended Use 219Practice Judging Context-rich Problems 222ii

Bibliography 229AppendicesAppendix A: Survey of Departments 235Appendix B: Context-Rich Mechanics Problems 239Appendix C: Context-Rich Electricity and Magnetism Problems 261Appendix D: Problem Solving Laboratories 271iii

1Chapter 1IntroductionIn this chapter: How to use this book. Course content and structure for Cooperative Problem Solving. Preparation to implement Cooperative Problem Solving (CPS). Steps needed to adapt CPS. What to expect after implementing CPS. Our “Laws of Instruction” and frequently used icons in this book.The most effective teaching method depends on the specific goals of acourse, the inclination of the instructor and the needs of the students,bound by the constraints imposed by the situation. There is no known"best" way to teach. Cooperative Problem Solving is one teaching tool that mayfit your situation. It is not the "magic bullet" that, by itself, will assure that all ofyour students achieve your goals for the course. It is, however, based on a solidresearch foundation from cognitive psychology, education, and physicseducation. We have over two decades of experience testing and refiningCooperative Problem Solving at it is used by many professors teachingthousands of students and different institutions. Cooperative Problem Solvingcan be used as the major focus of a course, or as a supplement in combinationwith other teaching tools.What is Cooperative Problem Solving (CPS)? This book is designed to answerthis question. First lets describe what you would see if you observed a 50-minute class engaged in Cooperative Problem Solving. As they walk into theclass, students sit in groups of three, facing each other and talking. Theinstructor begins class by talking about 5 minutes, setting the goal for thissession. For example, the instructor might remind the class that they have juststarted two-dimensional motion, that the problem they will solve today wasdesigned to help them understand the relationship between one-dimensional andtwo-dimensional motion, and they will have 35 minutes to solve the problem.The instructor also informs the students that at the end of 35 minutes, onemember from each group will be randomly selected to put part of their solutionon the board. The instructor then gives each group a sheet with the problem

2 Part 1:Teaching Physics Through Problem Solvingand all the fundamental equations they have studied in class to this point intime.The class is quiet for a few minutes asstudents read the problem, then there isa buzz of talking for the next 30minutes. No textbooks or notes areopen. Group members are talking andlistening to each other, and only onemember of each group is writing on apiece of paper. They mostly talk aboutwhat the problem is asking, how theobjects are moving, what they knowand don’t know, and what they need toassume, the meaning and application ofthe equations that they want to use, andthe next steps they should take in theirsolution. There are disagreements about what physics applies to the problemand what that physics means in this situation.The instructor circulates slowly through the room, observing and listening,diagnosing any difficulties the group is having solving the problem, andoccasionally interacting with the groups that the instructor judges need help.This pattern of listening to groups and short interventions continues for about30 minutes. At the end of that time, the instructor assigns one member fromeach group to draw a motion diagram and write the equations they used to solvethe problem on one of the boards on the walls. That member can ask for helpfrom the remaining group members as necessary. The instructor then tells theclass to examine what is on the board for a few minutes to determine thesimilarities and differences of this part of each group’s solution. The instructorthen leads a class discussion that highlights the similarities and differences andclarifies which are correct and which are incorrect. As the students are aboutleave class, the instructor hands out a complete solution to the problem.Almost everyone looks over the solutions. Some groups celebrate and othersgroan.The description above gives a image of a CPS class but does not include all thepreparations and scaffolding that go along with implementing CPS. In theremainder of this chapter we first outline how to use this book. This is followedby an outline of how CPS influences both content and course structure. Thenext section in this chapter includes a checklist for preparing to implement CPS,a brief outline of the steps needed to adapt CPS, and what to expect afterimplementing CPS. Finally, the last section of the chapter introduces some“laws of instruction” and icons that are frequently used in this book.

Introduction 3How To Use This BookThis book has four parts, described briefly below:Part I. Teaching Physics Through Problem solving. These six chaptersprovide background about the unsuccessful problem-solving strategies ofbeginning students, how context-rich problems and a problem-solvingframework help students engage in real problem solving, and why CooperativeProblem Solving (CPS) is a useful tool for teaching physics through problemsolving.Part 2. Using Cooperative Problem Solving. These four chapters describethe foundation of cooperative problem solving and provide detailed informationabout how to implement CPS for maximum effectiveness. This informationincludes what are appropriate group problems, how to manage groups, and thecourse structure and grading practices necessary to implements CPS. The lastchapter provides the research evidence that CPS improves both students’problem-solving skills and their conceptual understanding of physics.Part 3. Teaching a CPS Session. These three chapters provide informationabout how to prepare for a cooperative problem solving session, how toimplement the session, and how to monitor and intervene with groups as theyare solving problems.Part 4. Personalizing a Problem solving Framework and Problems.These three chapters provide some advanced techniques building upon knowledge fromprevious chapters and assuming that you have already tried using cooperative grouping.This part provides details on how to construct a problem-solving framework foryour students, how to write context-rich problems, and how to judge or adjustthe difficulty of a context-rich problem.The second page of each part summarizes the purpose and content of eachchapter of that part (pages12, 80, 134, and 164). You may want to read thesesummaries before you to help guide your reading.Random AccessWe have tried to write this book so that you can jump in anywhere you find anissue of interest. Of course you can also read it straight through. For toachieve that flexibility, each chapter contains frequent references to other relatedchapters. There is also some repetition of content to allow this type of randomaccess reading.Endnotes and ReferencesThe endnotes at the end of each chapter provide comments as well as referencesto seminal research papers or research reviews if you are interested in pursuing a

4 Part 1:Teaching Physics Through Problem Solvingsubject further. Since the research articles and reviews often apply to more thanone chapter, a bibliography is provided at the end of the book. To keep the listof references to a minimum, we have left out many important papers and booksthat can be found in the references of those cited.Course Content and Structure for Cooperative Problem SolvingTopics CoveredThe use of Cooperative Problem Solving (CPS) hasonly minor implications for how many topics you canincorporate into your course, and none, as far as weknow, for the order of those topics. Those decisionsmust be based on other factors such as the success ofyour students in meeting the goals of your course. CPS has been designed forcourses with a goal of having the majority of students reaching the appropriatelevel of a qualitative and quantitative understanding of all of the topics in thecourse. There is no method of instruction or course structure that can besuccessful if the course content is presented toorapidly for the average student in the class to have achance of understanding it.Class Size and Course StructureThe Cooperative Problem Solving techniquesdescribed in this book can be used under conditions that are less than ideal. Weteach at a large research-orientated state university with over 2000 students persemester taking physics courses taught in this manner. About one quarter ofthese students are in algebra-based courses and the rest are in calculus-basedcourses. Essentially all of these students take physics because their majorrequires it. The number of physics majors in any of the introductory courses isnegligible.The courses at our University have the very traditional structure of three largefifty-minute lectures per week, accompanied by smaller two-hour laboratoriesand fifty-minute discussion sections. Many different professors teach the lecturesections with the help graduate and undergraduate teaching assistants primarilyresponsible for the coordinated laboratories discussion sections. The teachingpersonnel changes each semester. This is not the best structure for CPSinstruction but it is still effective.At other institutions where Cooperative Problem Solving has been usedeffectively, teaching situations range from large classes with separated lecture,laboratories, and discussion sections, to smaller classes with all functionsinterwoven, taking place in a single room. Naturally each professor adapts CPSto fit his or her teaching style and situation. These techniques are being used atlarge research-oriented universities, comprehensive state universities, privateuniversities colleges, and community colleges.

Introduction 5Implementing Cooperative Problem SolvingIf you decide to teach a physics course using Cooperative Problem Solving(CPS), you need to be prepared to: Explicitly show students how to use the fundamental principles andconcepts of physics to solve problems (Chapters 4, 5 and 11). Use problems that require students to make decisions based on aworking knowledge of physics (Chapters 3 and 8). Have students solve problems while coached in groups, as well as solveproblems by themselves (Chapters 7, 12 and 13). Grade student problem solutions for communicating the application ofphysics while using sound problem solving techniques as well as thecorrectness of the solution. Grade students in a manner that emphasizesindividual achievement but does not discourage cooperation (Chapter9). Assume that all parts of the instructional process must be repeated everytime a new topic is introduced.Of course, "the devil is in the details." Successful implementation requiresattention to the learning process as well as the teaching process. In designingCooperative Problem Solving we tried many things that did not work, althoughthey seemed to be reasonable extensions of fundamental learning research orcommon sense experience. Even techniques that work well with one class maynot give as large an effect in another type of class. This is not surprisingbecause the learning process is complex. As with any complex system, theparameters that influence learning must be tuned for each particular set ofconstraints to achieve large effects. This book describes the sensitiveparameters that we have found in our research and development work. Theymust be matched to the goals and constraints of your situation.Overview of Adapting Cooperative Problem SolvingThere is no point in you making exactly the same mistakes we did in gettingCooperative Problem Solving to work. Based on the experience of faculty atother institutions that have adapted these techniques to their situations, wesuggest that you begin by following as much of our prescription as possible inyour situation. After you have tried it once, you will have a good idea of what tochange to make it work better for you.The following is an outline of what to do. The details and rationale are given inthe remainder of this book.Step Determine the most important goals for studentlearning in your class. For example, the goals below arethe top goals from a survey of the engineering faculty

6 Part 1:Teaching Physics Through Problem Solvingthat require their students to take introductory physics at the University ofMinnesota. See Appendix A for a copy of the survey. Know the basic principles underlying all physics. Be able to solve problems using general qualitative logical reasoningwithin the context of physics. Be able to solve problems using general quantitative problem-solvingskills within the context of physics. Be able to apply the physics topics covered to new situations notexplicitly taught by the course. Use with confidence the physics topics covered.Step Get or write context-rich problems that require students to: Directly use the physics concepts you want to teach; Directly address the goals of your course; and Practice their weakest problem solving skills.Appendices B and C contain examples of context-rich problems inmechanics and electromagnetism. See Chapter 3 for a description ofcontext-rich problems, Chapter 15 for how to write context-rich problems,and Chapter 16 for how to judge and adjust the difficulty of those problems.Step Adopt a research based, problem-solving framework that you wantyour students to use to solve problems. Make every step of the processexplicit to the students. Always demonstrate the same logical and complete problem-solvingprocess throughout the course no matter what the topic. Explain all the decisions necessary to solve the problem. Show every step, no matter how small, to arrive at the solution. Hand out or have on the web examples of complete solutions toproblems showing how you expect students to communicate theirthought process in problem solutions. Allow each student to make their own reasonable variations of theframework for their solutions.See Chapter 6 for additional information about demonstrating a problemsolvingframework. Chapter 4 provides a description and example of aresearch based problem-solving framework, and Chapter 5 providesexamples of complete problem solutions using the framework. Chapter 14describes how to personalize a framework to match your preferences andthe needs of your students.

Introduction 7Step Require that your students practice solvingproblems in small groups while you or anotherinstructor provide timely guidance. Use cooperative groups of three, or at mostfour. Structure the groups so that they really worktogether on the solution. Use a problem appropriate for group work.Make sure no notes or books are available to students while they solveproblems in groups. Grade the results of these group problems, only occasionally, for alogical and complete solution using correct physics reasonablycommunicated.See Chapter 7 for a description of how Cooperative Problem Solving isdifferent from having students work in groups, and Chapters 8 - 9 forappropriate group problems, structuring and managing groups, and grading.See also Chapters 11-13 for how to prepare for and teach a CooperativeProblem Solving session.Step Encourage students to solve homework problems by themselves usingcorrect physics communicated in a logical and complete manner. If homework is graded, make sure the gradedepends on logical and complete solutions, notjust a correct answer. If you cannot grade homework in this manner (wedon’t have the resources) make at least oneproblem on your tests an obvious modification ofa homework problem. Point this out to your students. A week or twobefore each test, give students a sample test to work on at home.See Chapter 9 for how to grade problem-solving performance.Step Give the same type of problems on your examinations that yourstudents solve in their groups. Grade them based on the student behavioryou wish to encourage Grade on an absolute scale to encourage student cooperation. Grade for well-communicated problem solutions presented in anorganized and logical manner. Give enough time to actually solve problems based on makingdecisions about physics and writing complete solutions.See Chapters 9 for how to grade problem-solving performance.

8 Part 1:Teaching Physics Through Problem SolvingAfter becoming comfortable with the basics of Cooperative Problem Solving,you can improve student achievement by using all of your course resources toaddress your goals. For example, all demonstrations can be presented asexamples of problem solving. In addition, laboratories can be structured so thatthey give students practice in problem solving that can be checked bymeasurement. See Appendix D or visit our website(http://groups.physics.umn.edu/physed/research.html) for more informationabout problem solving labs.What to Expect After Implementation of CPSIf the implementation of Cooperative Problem Solving (CPS) is going well, byabout half way through your term you should notice some changes in your class.When you visit discussion sections you will hear students talking to each otherabout physics concepts, and beginning to use the language of physics. Studentproblem solutions on exams will look neater and more organized and thus easierto grade. By the end of the term you will find that student drop-out rates, itthey were high, will have decreased. Student conceptual knowledge as measuredby objective exams such as the Force Concept Inventory will have improved.The problem-solving performance of your students will also improve. Yourstudents will be more willing to tackle new and unusual problems. They willtend to begin problems by thinking about what physics to apply. Typically, youcannot expect things to go smoothly until the second time you have taught acourse after you implement any significant change.Of course you will not be 100% successful and not all your students will likeCPS. Judge your success by comparing to results of previous classes. Researchshows that in traditional classes, about 20% of the students show significantimprovement. Our goal is to reach 2/3 of the students 2/3 of the time. In thebeginning, try not to worry if one of five groups is dysfunctional. That's an 80%success rate! Spend most of the time on most of the students. Next timearound you can try to tweak the course structure to reach the others. Makeincremental changes. Try to improve your results just a little bit each year.Students are usually the most conservative element in the educational process.They may not like how things are now taught, but they resist any changes. If thecourse is noticeably different than it was the previous year, students will blametheir difficulties on those changes. Be patient and supportive of your studentsat the beginning. You must believe that the changes are for the better. If youdon't have confidence in what you are doing, then you can't expect them to.Naturally you will meet the most resistance from students that have beensuccessful previously. Human beings do not like to change, especially if theyhave been successful. On the other hand, students who were expecting to havetrouble will be immediately grateful. As long as you are firm and positive andthe students are more successful than they imagined, they will be positive aboutthe changes at the end of the course. Interestingly we have found that thestrongest support for Cooperative Problem Solving comes from some of the

Introduction 9students who resisted the most at the beginning. In the hands of a very goodstudent this is a very powerful tool and they recognize it.Our Laws of Instruction and Other Frequently Used IconsTo guide the actual implementation of a course design, we have invented, onlyhalf seriously, four "Laws of Instruction" in analogy with the "Laws ofThermodynamics." In the same spirit as classical thermodynamics, these "laws"describe robust empirical observations that are based on the current state ofknowledge of human behavior and learning. 1 The overriding principle of humanbehavior addressed by the laws is that most human beings do not like to changetheir behavior. As with thermodynamics, our “laws” are statistical in nature --you will certainly know of specific counter examples.Our Laws of Instruction are described below.If you don't grade for it, students won't do it.It would be wonderful if we lived in a world in which students were intrinsicallymotivated to learn new things, and our physics class was the only class studentswere taking. But it just isn’t so. Humans expend the minimum energy necessaryto survive. Students expend the minimum energy necessary to get what theyconsider a “good” grade.Doing something once is not enough.The most effective way for humans to learn any complex skill is apprenticeship.For example, a new apprentice would learn tailoring in a busy tailor shop, wherehe or she is surrounded both by master tailors and other apprentices, all engagedin the practice of tailoring at varying levels of expertise. Masters teachapprentices through a combination of activities called modeling, scaffolding,coaching and fading. 2 Repetition of these activities is essential.Modeling. The apprentice repeatedly observes the master demonstrating (ormodeling) the target process, which usually involves many different but relatedsub-skills. This observation allows the apprentice to build a mental model ofthe processes required to accomplish the task.Scaffolding. Scaffolding is structure that supports the learning of theapprentice. Scaffolding can include a compelling task or problem, templates andguides, practice tasks, and collections of related resources for the apprentice.Usually scaffolding is removed as soon as possible but may need to bereintroduced later for a new context.Coaching. The apprentice then attempts to execute each process with guidanceand help from the master (i.e., coaching). A key aspect of coaching is thesupport, in the form of reminders or help that guides the apprentice toapproximate the execution of the entire complex sequence of skills, in their ownway. The interaction with other learners provides the apprentice with instantfeedback through peer coaching and a calibration of progress, helping focus the

10 Part 1:Teaching Physics Through Problem Solvingeffort needed for individual improvement.Fading. Once the apprentice has a grasp of the entire process, the masterreduces the scaffolding (i.e., fading), providing only limited hints, refinements,and feedback to the apprentice, who practices by successively approximatingsmooth execution of the entire process. The interplay between observation,scaffolding, peer interactions, expert coaching, and increasingly independentpractice helps the apprentice develop self-monitoring and correction skills andintegrate those skills with other knowledge to advance toward expertise.Problem solving is a complex mental skill. Like learning a complex physicalskill, students learn physics problem solving best in an environment in whichproblem solving is modeled, the process is scaffolded, they are coached in theprocess, and the structure and coaching fades as the students become betterproblem solvers. All of this should happen in what is called an environment ofexpert practice in which the student should be able to answer the followingthree questions at any time in the course:1. Why is whatever we are now learning important?2. How is it used?3. How is it related to what I already know?[See Chapter 6 for a more detailed description of cognitive apprenticeship.]Don't change course in midstream; structure early then graduallyreduce the structure.Humans are very resistant to change. It is easier on both instructors andstudents to start with what may seem a rigid structure (e.g., a problem-solvingframework, roles for working in groups), then fade gradually as the structure isno longer needed. It is almost impossible to impose a structure in the middle ofthe course, after you discover that students need it.Make it easier for students to do what you want them to do andmore difficult to do what you don’t want.Humans will persist in apreviously successfulbehavior until it is no longerviable for survival. Learninga new way of thinking is adifficult, time consuming,and frustrating process, likeclimbing a steep mountain.For most students, expertlikeproblem solving is anew way of thinking.Students will try to runaround this mountain byusing their unsuccessful problem-solving strategies (e.g., plug-and-chug andpattern-matching, see Chapter 2), especially if they seem to almost work.

Introduction 11For students to learn physics through problem solving, it is not enough tomodel problem solving, scaffold and coach (provide ladders up the mountain),and gradually fade the support. You must also supply barriers (fences) to makeit obvious to students that their novice problem-solving strategies areunsuccessful.IconsIn addition to the icons for the four Laws of Education, we have used thefollowing icons to highlight certain information within each chapter.RememberThis icon is a reminder of related ideas, often described in other chapters of thebook, that are helpful in understanding the current topic.WarningA warning icon denotes a research-based recommendation does not match atraditional, common sense teaching practice.TipTeaching tips are practices that has been found useful in implementingCooperative Problem Solving (CPS).Endnotes1For a summary of the research on how people learn, see Bransford, J.D., Brown, A.L., andCocking, R.R. (Eds) (2000), How people learn: Brain, mind, experience, and school, Washington DC,National Academy Press. For a description of seven principles of learning based on thisresearch, see National Research Council. (2002), Learning and understanding: Improving advancedstudy of mathematics and science in U.S. high schools, Washington, D.C., National Academy Press.Research-based principles of learning are also described in Reddish, E.F. (2003), Teachingphysics with the physics suite, New York NY, John Wiley & Sons Inc.2See, for example: Collins, A., Brown, J.S. & Newman, S.E. (1989), Cognitive apprenticeship:Teaching the crafts of reading, writing and mathematics, in Knowing, Learning, and Instruction:Essays in Honor of Robert Glaser edited by L.B. Resnick, Hillsdale NJ, Lawrence Erlbaum, pp.453 – 494; and Brown, J. S., Collins, A., & Duguid, P. (1989), Situated cognition and theculture of learning, Educational Researcher, 18(1), 32-42.

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13PartHow to Avoid Solving Problems(unknown internet author)

14In this part . . .You can explore some of the underlying justification for the techniques wedescribe in this book. If you just want to know “how to do it,” you can skipthis part.Chapter 2 describes why most instructors teach physics through problemsolving, the difference between student and expert problem solving, and whatstudents typically learn.Chapter 3 gives two applications of our 3rd Law of Instruction. The first is thedesign of context-rich problems and the second is the control of the equationsthat students use to solve those problems.Chapter 4 describes the features of a general problem-solving framework, andgives an example of a problem-solving framework used in introductory physicscourses. Included in the example is a running commentary explaining thepurpose and rationale for each part of the solution using this framework.Chapter 5 describes three different ways of presenting a problem solvingframework to students – flow charts of problem-solving decisions, answersheets for problem solutions, and example problem solutions using the answersheets.Chapter 6 presents the rationale for Cooperative Problem Solving – whatdifficulty does it address? We also present a theoretical framework for thestructure of these techniques.

15Chapter 2Connecting Students, Physics,and Problem SolvingIn this chapter: Why physics faculty teach physics through problem solving Knowledge organization – expert and novice networks of ideas. Students’ novice problem-solving strategies, the plug-and-chug and pattern-matchingstrategies. Who benefits from traditional teachingStudents come to us with well-developed knowledge, including their personalphysics ideas 1 , and expectations about how and what they will learn in aphysics class. 2 Decades of research have shown that students’ personal ideasabout physics often do not match established physics concepts. For example, about80% of students entering our calculus-based introductory physics course think thatduring a collision between a large truck and a small car, the force of the truck onthe car is larger than the force of the car on the truck. Student’s personal ideas thatdo not match physics concepts are often called preconceptions, naive conceptions,alternative conceptions or misconceptions.Figure 2.1. Model of neurons firing in the brain

16 Part 1:Teaching Physics Through Problem SolvingFigure 2.2. Example of a network of a student’s personal ideasabout the Newton’s third law of motion.Although many details remain to be determined, there is a consistent model ofcognition that has emerged from neuroscience and cognitive science. In thismodel, elements of knowledge are networks of connected neurons. When someoneuses the knowledge represented by a particular network, the neurons in thatnetwork are activated – increase in their firing rate (see Figure 2.1). Networks arisefrom building associations among neurons through synapse growth. These neuralconnections are built when different neurons are activated at approximately thesame time. This process is summarized by the slogan “neurons that fire together,wire together” attributed to Hebb. [reference Hebb, D (1949). The organization ofbehavior. New York: Wiley.]Knowledge stored in long-term memory is the way those neurons are linked, andthe way those linked structures are activated. 3For example, a student’s knowledge of the Newton’s third law can berepresented as a network of these connections of ideas which are themselvesnetworks of neural connections, as illustrated in Figure 2.2. 4 Some of thisknowledge might be relatively isolated and some interwoven with that of otheractivities, such as driving a car or playing basketball. The biological process oflearning is complex, requiring the establishment and deletion of connectionsthat differ in detail for every individual. Because everyone filters theirperceptions through their knowledge network, instructional input by itself, isnot sufficient for a majority of students to learn physics or any other field.Neither clear explanations, nor dramatic demonstrations, nor laboratory

Chapter 2: Connecting Students, Physics, and Problem Solving 17activities are enough.In this chapter we first describe why physics is traditionally taught throughproblem solving and what students learn. The second section describes whatstudents usually learn when they solve physics problems.Why Teach Physics Through Problem Solving?TraditionWhen experts solve a physics problem they link fundamental physics principlesand concepts, such as Newton’s laws of motion or the conservation of energy,to their knowledge of a physical situation. 5 For example when throwing a ballover short distances, experts know that the drag force can be ignored; theacceleration in the vertical direction is a known constant; and there is noacceleration in the horizontal direction. Using this knowledge they can predictthe motion of the ball through logic, including mathematics, based on the initialvelocity of the ball. An example of part of the network of knowledge of anexpert, such as yourself, is shown in Figure 2.3.Solving problems in this manner requires a deep understanding of fundamentalphysics concepts, including their utility in particular situations. A correctsolution embodies both correct physics concepts and their properinterconnection to other ideas that are related to the physical situation of theproblem. This is what we want our students to learn.In addition to our own traditions, there are external reasons to teach physicsthrough problem solving.Other Majors in Our CoursesSurveys of university faculties show that gaining experience in problem solving isone of the primary reasons that other majors require their students to takeintroductory physics. (An example of the survey we used is included in AppendixA). This is as true for the departments that require algebra-based physics as it is forthe science and engineering departments that require calculus-based physics.Employment of Physics MajorsSurveys of physics majors, after they have graduated andare functioning in jobs, shows that problem solving is theprimary skill from physics that they use, as shown inFigure 2.4. It is interesting to note that interpersonalskills, including teamwork skills, is the second skill usedmost often by graduates in their jobs.

18 Part 1:Teaching Physics Through Problem SolvingWhat do students typically learn through problem solving?We all have students that are reasonably successful at getting the correct answer tosome of the end-of-chapter problems in the text, but have difficulty solving themore complex problems in the text and on exams. Many beginning studentsattempt to answer physics problems in a manner that we do not recognize asproblem solving.Their solution strategy is to connect the known quantities in a problem with asimilar quantities in an equation, and then perform mathematical manipulationsuntil the answer appears. A typical solution of such a student is shown in Figure2.5. This strategy, known as “plug and chug”, promotes learning of physics bymemorizing equations and practicing the mathematical manipulation of thoseequations .6 Most students employing this strategy do not do well in a universityphysics course. They complain that there is too much to learn. From their point ofview they are correct.Most students who survive their course haveusually generalized their strategy from trying tolearn the specific formula to solve eachproblem, to trying to learn the pattern of equationsto solve different classes of problems. Each class ofproblems is characterized by a literal feature ofthe problem, such as the specific action of theobjects involved. These students try toremember solution patterns, usually fromexample solutions in the textbook or thelecture, and attempt to force their solution tofit the pattern. 7For example, a student might try to remember the pattern of mathematicalsteps for solving problems involving “objects sliding down an incline plane” or“objects moving in a circle.” Pattern-matching students often remark that theycan’t figure out how to begin a problem. No matter how many worked outsolutions the instructor gives, they are always asking for more workedexamples.The student solution in Figure 2.6 illustrates this “pattern-matching” strategy.This student tried to match the solution pattern for “objects launchedupwards” to this situation. As the solution illustrates, a small variation in theproblem can derail such students -- they are not able to generalize a problemtype to similar problems with different objects, events, or constraints. 8 Even ifthe pattern-matching behavior results in the correct answer, the process itself isnot very effective for learning either physics concepts or their applicationthrough problem solving. Pattern matching leads to an incoherent andfragmented network of knowledge -- one that is not organized aroundfundamental concepts. For example, compare the network of knowledge of anexpert (Figure 2.3) with that of a pattern-matching student shown in Figure 2.7.

Chapter 2: Connecting Students, Physics, and Problem Solving 19Figure 2.3. Example of a small part* of the expert network ofknowledge for solving a novel projectile-motion problem.* Not shown are the links to many experiences with projectile motion and the links to theconservation of energy.

x y at 2 t 2 500 m20 Part 1:Teaching Physics Through Problem SolvingFigure 2.4. Survey of employees with physics bachelor’s degreesThe plug-and-chug and pattern-matching behavior of students is documented byresearch into what are called is called “novice” problem-solving strategies. Evensuccessful students who can answer the end-of-chapter textbook problems typicallydo not have a grasp of the basic physics concepts involved. 9 In other words,research shows that we, the physics faculty, have been correct all along. Studentswho do poorly on our examination problems really do not understand the basicconcepts of physics.Figure 2.5. Example of a plug-and-chug novice problem-solving strategy (takenfrom an actual student test solution in an algebra based physics class)Problem: A boulder, rolling horizontally at a constant speed, goes off the edge of a500-ft cliff. How fast would it need to be rolling to hit 100 feet from the base of thecliff?500 m100 m509.9 mtan oppadjtan 1 100500 11.3500 2 100 2 260000 509.9mx y v o t 1 2 at2t x v500 9.8 m/s 2

Chapter 2: Connecting Students, Physics, and Problem Solving 21v atv = x txt att 2 (9.8 m/s 2 )(500 m)xt 2 51.0 sa t2 t 7 sec.71.4m/s =11x x o v o t 1 2 at2x x otx x o v o t 1 2 at2a = g = 9.8 m/s 1 2 gt v o 0 -500 m7 sx x o v o 1t 2 gt 1 2 9.8 m s 2 7 s v otan 11 v ox 71.4 m /sV ox71.4 34.3 37.13 m/sv ox 13 .9 m / sThe boulder would have to roll at 13.9 m/s.xt v500 m7 sec v yv y 71.4 m/sFigure 2.6. An example of a novice “pattern-matching” problem-solving strategy (this is an actualstudent solution on a test in an algebra based physics class)Problem: A boulder, rolling horizontally at a constant speed, goes off the edge of a500-ft cliff. How fast would it need to be rolling to hit 100 feet from the base of thecliff?500 mcv i ?x 500 my = 100 mt =?500 2 100 2 510 mc = 510 m100 mx v i t 1 xx vcost 12 gt22 gt2 x 0x v cos 1 y y vsint 12 gt22 gt2 y 0y v sin 1 vcost 12 gt2 2 gt2 x vsint 1 2 gt2 yvcost vsin xyvt cos - sin 100 500 400 m

22 Part 1:Teaching Physics Through Problem SolvingFigure 2.7. Example of a novice student’s pattern-matching networkof knowledge for solving projectile motion problemsStudents often complain that such problems are not clear or tricky. Over timemany instructors tend to modify their test problems so that students can answer theproblems using their novice problem-solving strategies. Then they can get the rightanswer and still not understand either the physics concepts or the process ofproblem solving. The very human response to minimizing student frustration oftendrives instructors in the direction of building questions which minimize learning.Who benefits?Real problem solving is an additional teaching tool that requires each student toexamine his or her own mental connections. This process can be difficult, timeconsuming, and frustrating. Most human beings work very hard to avoid engagingin this process, as illustrated in the problem-solving flow chart that circulatedaround the internet some years ago (shown on cover page to Part 1, page 11). Thevast majority of students entering our introductory physics courses do not engagein real problem solving. Without instruction in problem solving, those who survivetend to use a pattern-matching strategy while unsuccessful students engage in theplug-and-chug strategy.In the classroom, it is useful to remember the old practice of miners using canariesto detect poisonous gasses underground. Unsuccessful students are more sensitive

Chapter 2: Connecting Students, Physics, and Problem Solving 23to weak teaching practices, so like the canaries in mines, it is easy to see theirfailure. If a significant number of students are dropping out or failing a course,then it is probable that the successful students who, like the miners survive, are alsobeing harmed.Good teaching practices most obviously benefit the typically unsuccessful students,but they equally benefit the best students. In the next three chapters in Part 1, wedescribe some teaching practices that benefit both the typically successful andunsuccessful students.Endnotes12There is a vast literature on college students’ personal physics ideas -- also calledalternative conceptions, preconceptions, naive conceptions, and misconceptions.A sampling of articles include: Trowbridge, D.E. & McDermott, L. C. (1981),Investigation of student understanding of the concept of acceleration in onedimension, American Journal of Physics, 49(3), 242-253; Clement, J. (1982),Students' preconceptions in introductory mechanics, American Journal of Physics,50(1), 66-71; McCloskey, M. (1983), Naive theories about motion, in MentalModels, D.S. Gentner and A. Stevens (Eds), Hillsdale, New York: LawrenceElbraum; McDermott, L.C. (1984), Research on conceptual understanding inmechanics, Physics Today, 37(7), 24-32; Goldberg, F.M. & McDermott, L.C.(1987), An investigation of student understanding of the real image formed by aconverging lens or concave mirror, American Journal of Physics, 55(2), 108 –11;Cohen, R., Eylon, B. & Ganiel, U. (1983), Potential difference and current insimple electric circuits, American Journal of Physics, 51, 407 - 412; Guth, J. (1995),An in-depth study of two individual students’ understanding of electric andmagnetic fields, Research in Science Education, 25(4), 479-490; Furio, C. &Guisasola, J. (1998), Difficulties in learning the concept of electric field, ScienceEducation, 82(4), 511-26; Loverude, M.E., Kautz, C.H., & Heron, P.R.L. (2002),Student understanding of the first law of thermodynamics: Relating work to theadiabatic compression of an ideal gas, American Journal of Physics, 70(2), 137; andKautz, C.H., Heron, P.R.L., Shaffer, P.S. and McDermott, L.C. (2005), Studentunderstanding of the ideal gas law, Part II: A microscopic perspective, AmericanJournal of Physics, 73(11), 1064-1071. For an overview of some of these studies,see McDermott, L.C. & Redish, E.F. (1999), Resource letter PER-1: Physicseducation research, American Journal of Physics, 67, 755-767.There is a growing research base about students’ expectations or beliefs about thenature of physics knowledge and how to learn physics that affect student learningin a class. See, for example: Redish, E.F., Saul, J.M., & Steinberg, R.N. (1998),Student expectations in introductory physics, American Journal of Physics, 66, 212-224; Hammer, D. (2000), Student resources for learning introductory physics,American Journal of Physics, Physics Education Research Supplement, 68(S1), S52-S59; Elby, A., and Hammer, D. (2001). On the substance of a sophisticated

24 Part 1:Teaching Physics Through Problem Solvingepistemology. Science Education, 85, 554-567; Hammer, D. (1994), Epistemologicalbeliefs in introductory physics. Cognition and Instruction, 12 (2), 151-183; andPerkins, K. Adams, W. K., Finkelstein, N. D., Pollock, J. S., & Wieman, C. E.(2005), Correlating student attitudes with student learning using the ColoradoLearning Attitudes about Science Survey, Proceedings of the 2004 Physics EducationResearch Conference, Rochester, NY: PERC Publishing3456Directly from an article by Jonathan Tuminaro and Edward Redish (2007),Elements of a cognitive model of physics problem solving: Epistemic games,Physical Review ST Physics Education Research, 3, 020101. There are many booksabout the emerging models of cognition. See, for example, Anderson, J.R.(2004), Cognitive science and its implications, 6th Ed, Worth Publishing; Fuster, J.(1999), Memory in the cerebral cortex: An empirical approach to neural networks in thehuman and nonhuman primate, MIT Press; and Fuster, J. (2003), Cortex and mind:Unifying cognition, Oxford University Press.In this brief presentation, we are using the more familiar term, ideas, rather thanthe smaller elements of knowledge with which students think. For a briefdiscussion of these elements of knowledge, see the introduction to an article byJonathan Tuminaro and Edward Redish (2007), Elements of a cognitive model ofphysics problem solving: Epistemic games, Physical Review ST Physics EducationResearch, 3, 020101.For some seminal papers about how experts and novice solve problems, see Chi,M.T.H., Feltovich, P. J., & Glaser, R. (1981), Categorization and representationof physics problems by experts and novices, Cognitive Science, 5, 121-152; Eylon, B.& Reif, R (1984), Effects of knowledge organization on task performance,Cognition and Instruction, 1(1), 5-44; and Larkin, J., McDermott, J., Simon, D. andSimon, H. (1980), Expert and novice performance in solving physics problems,Science, 208, 1335-1342.Schoenfeld, A. H. (1983). Beyond the purely cognitive: Belief systems, socialcognitions, and meta-cognitions as driving forces in intellectual performance,Cognitive Science, 8, 173-1907Of course, there are more than two novice strategies for solving problems.Different researchers use different research methodologies (e.g., videotapingindividual students thinking aloud while solving a problem with structured orsemi-structured questions by the interviewer; videotaping groups of studentssolving homework problems), students from classes with different pedagogies,and describe the novice strategies in different ways. See, for example: Tuminaro,J & Redish E.F. (2007), Elements of a cognitive model of physics problemsolving: Epistemic games, Physical Review ST Physics Education Research, 3, 020101;Walsh, L.N., Howard, R.G., and Bowe, B. (2007), Phenomenographic study ofstudents’ problem solving approaches in physics, Physical Review ST PhysicsEducation Research, 3, 020108.8See, for example, Chi, M., Feltovich, P.J., & Glaser, R. (1981), Categorization

Chapter 2: Connecting Students, Physics, and Problem Solving 25and representation of physics problems by experts and novices, Cognitive Science,5, 121-152; Reif, F., & Heller, J.I (1982), Knowledge structure and problemsolving in physics, Educational Psychologist, 17, 102-127; Ferguson-Hessler,M.G.M., & de Jong, T. (1987), On the quality of knowledge in the field ofelectricity and magnetism, American Journal of Physics, 55(6), 492-497; Elio, R. &Scharf, P.B. (1990), Modeling novice-to-expert shifts in problem solving strategyand knowledge organization, Cognitive Science, 14(4), 579-639; and Zajchowski, R.,& Martin, J. (1993), Differences in the problem solving of stronger and weakernovices in physics: Knowledge, strategies, or knowledge structure? Journal ofResearch in Science Teaching, 30, 459-470.9For reviews of expert-novice research, see Maloney, D.P. (1994), Research onproblem solving in physics. In D.L. Gabel (Ed.), Handbook of Research in ScienceTeaching and Learning, (pp. 327-354), NY, Macmillan. See also, Hsu, L., Brewe,E., Foster, T. M., & Harper, K. A. (2004), Resource letter RPS-1: Research inproblem solving. American Journal of Physics, 72(9), 1147-1156.

26 Part 1:Teaching Physics Through Problem Solving

27Chapter 3Combating Problem SolvingThat Avoids ContentIn this chapter: How context-rich problems help students engage in real problem solving The relationship between students’ problem-solving difficulties and the design ofcontext-rich problems How to discourage students from indiscriminate formula memorization.Real problem solving has been described asthe process of arriving at a solution whenyou don’t initially know what to do. 1 Thismeans that problem solving requires makingdecisions. Making decisions involves the connectionand application of different types of knowledge (e.g.,concepts, facts, and procedures) to construct asatisfactory solution. Because problem solving usesgeneral-purpose tools, such as fundamental physicsconcepts, it should be a good vehicle for assisting inthe learning of those concepts.Of course problem solving is often a difficult, time consuming, and frustratingprocess -- like climbing a steep mountain. Most students either give up or avoidthis mountain by using their novice strategies such as plug-and-chug or patternmatching.Consequently, our Third Law of Education needs to be applied, asillustrated in Figure 3.1. [See the Introduction, pages 9-10, for a completedescription of our Laws of Education.] The Third Law is:Make it easier for students to do what you wantthem to do, and more difficult to do what youdon’t want.In this chapter and chapters 4 and 5, we describe applications of this Law ofEducation that helps all students become more competent problem solvers.Two applications of the Third Law are described in this chapter. The first is thedesign of context-rich problems, and the second is the restriction of the“formulas” allowed on exams.

28 Part 1:Teaching Physics Through Problem SolvingFigure 3.1. Illustration of the Third Law of Education: Make it easier for students to do what youwant them to do (ladders) and more difficult to do what you don’t want (fences).Design of Context-rich Problems: How They Help Students Engagein Real Problem SolvingWe have developed a type of question, context-rich problems, that are designedto encourage students to engage in real problem solving while discouragingstudents’ natural tendency use novice problem-solving strategies (see Chapter 2for novice problem-solving strategies). The goal of this type of problem is togive students practice incorporating physics into their existing knowledgenetwork. If students understand the basic physics involved, the problem shouldbe easy to comprehend and straightforward to solve. If students do not havethat understanding, they should not be able to make progress toward a solutionat the point that the physics concept arises.In other words, it should be obvious to the student, as well as the instructor,where the difficulty lies so that they can examine their own connections to thatphysics and get help if necessary. The following is an example of such aproblem that might be given early in an introductory physics course that beginswith kinematics:

Chapter 3: Combating Problem Solving That Avoids Physics 29Figure 3.2. The Traffic Accident context-rich problemYou have a summer job with an insurance company and are helping to investigate atragic accident. At the scene, you see a road running straight down a hill that is at10° to the horizontal. At the bottom of the hill, the road widens into a small, levelparking lot overlooking a cliff. The cliff has a vertical drop of 400 feet to thehorizontal ground below where a car is wrecked 30 feet from the base of the cliff. Awitness claims that the car was parked on the hill and began coasting down the roadtaking about 3 seconds to get down the hill. Your boss drops a stone from the edgeof the cliff and, from the sound of it hitting the ground below, determines that ittakes 5.0 seconds to fall to the bottom. You are told to calculate the car's averageacceleration coming down the hill based on the statement of the witness and theother facts in the case. Obviously, your boss suspects foul play.Some of the features of this context-rich problem are explained below. Thesefeatures are common to all context-rich problems. Some features are designedto encourage students to incorporate physics principles and problem solvingpractices into their knowledge network by making decisions based on theirexisting ideas.(ladders). Other features are designed to discouraging students’natural tendency use novice problem-solving strategies (fences). Above all, thistype of problem requires the student to make and link decisions in a logical andorganized manner, the hallmark of real problem solving.Feature 1. It is difficult to use a few equationsand plug in numbers to get an answer.The student “knows” that acceleration is velocity dividedby time, but no velocity is given and there are two differenttimes in the Traffic Accident problem. The student also“knows” that velocity is distance divided by time, so onetime is to get the velocity and the other is to get theacceleration. Unfortunately for the student, there are twodistances in the problem. Which one should be used?There is also an angle given. Does the student need tomultiply something by a sine or cosine?Feature 2. It is difficult to find a matchingsolution pattern to get an answer.Going down the hill at a known angle looks like an“inclined plane problem.” Is the acceleration justg sin? But what about the other numbers in the problem?When the car goes off the cliff, this is a “projectileproblem.” You can calculate an acceleration from thedistance (which one?) and the time the car falls. Whywould that be the acceleration down the hill?

30 Part 1:Teaching Physics Through Problem SolvingFeature 3. It is difficult to solve the problem without firstanalyzing the problem situation.It is difficult to understand what is going on in this problem without drawing apicture and designating the important quantities on that picture. Making the situation as real as possible,including a plausible motivation, helpsstudents in the visualization process. (ladder) Making the student the primary actor in theproblem also helps the visualization process.This also avoids gender and ethnic biases thatcan inhibit learning. The other actors in theproblem are as generic as possible, so the student’s visualization is nothampered by unfamiliar names or relationships. (ladder) Students are forced to practice visualization because no picture is givento them. What are the velocity and acceleration of the car at interestingpositions in its motion? What are those positions? (fence) The visualization of a realistic situation gives the student practiceconnecting “physics knowledge” to other parts of the student’sknowledge structure. This makes the physics more accessible, and somore easily applied to other situations. What does a car going off acliff have to do with dropping a stone? Does physics really apply toreconstructing accidents? (ladder) In real situations, assumptions must always be made. What arereasonable assumptions? What is the physics that justifies makingthose assumptions? Can friction be ignored? Where? Is theacceleration down the hill constant? Do you care? This gives studentspractice in idealization to get at the essential physics behind complexsituations. (ladder)Feature 4. Physics cues, such as “inclined plane”, “starting fromrest”, or “projectile motion”, are avoided.Avoiding physics cues not only makes it difficult forstudents do pattern matching, it encourages students tobuild the connections between physics and their existingknowledge structure. Using common words helps students practiceconnecting physics knowledge to other things theyknow. This makes the physics more capable ofbeing applied to other situations. Here the car goesdown a hill and begins by being parked. (ladder) Physics cues tend to set students thinking along a predetermined path.

Chapter 3: Combating Problem Solving That Avoids Physics 31They do not require students to examine their physics concepts todetermine which are applicable. Here the student must apply physicsknowledge to decide on where the acceleration of the car is constantand where the velocity, or a component of the velocity of the car isconstant. (fence)Feature 5. Logical analysis using fundamental concepts isreinforced.Logical analysis is reinforced because there is no obvious path from theinformation given to the desired answer. Each student must construct thatpath incrementally. Using a logical analysis helps to determine which information isrelevant and which is not. The extra information is not put in toconfuse the student but to force informed decision making. It isinformation that they would likely have in that situation and could berelevant. It might show that there is a choice between two equally validsolution paths or actually be irrelevant so it would only be used if thestudent has incorrect or fragile physics knowledge. (ladder and fence) The answer to the problem can be arrived at in a straightforwardmanner after a logical analysis using the most fundamental physicsconcepts, in this case the definition of average acceleration and averagevelocity as well as the connection between average and instantaneousvelocity for constant velocity. (ladder) A logical analysis is necessary because this question cannot be answeredin one step. (fence)If you think such context-rich problems are too difficult for most of yourstudents at the beginning of your class, you are right. This problem wasdesigned for either: (1) an instructor demonstration of a logical and coherentframework for solving this problem, or (2) for a group of students to use theframework to co-construct a solution. In the next chapters we describeproblems-solving frameworks and the reason for cooperative problem solving.Relationship Between Students’ Problem-solving Difficulties andthe Design of Context-rich ProblemsFor contrast, here is the same traffic-accident problem(Figure 3.2) with only the “essentials.”A block starts from rest and accelerates for 3.0seconds. It then goes 30 ft. in 5.0 seconds at aconstant velocity.a. What was the final velocity of the block?b. What was the average acceleration of the block?

32 Part 1:Teaching Physics Through Problem SolvingFigure 3.3. Table of student difficulties and context-rich problem design featuresStudent Difficulty Symptom Design FeatureVisualizing aphysical situationConnecting physicsto realityRecognizing theunderlyingfundamental physicsof a situation.Idealization.Application offundamental concepts.Integrating knowledgeinto a coherentconceptual framework.Executing alogical analysis.Over relianceon pattern matchingLack ofmathematical rigor.Gender orethnic bias.Physically impossible results.No pictures or diagramsdrawn.Physically impossible results.Difficulty in applyingknowledge to slightly differentsituations.Difficulty applying knowledgeconsistently, even within asingle situation.Difficulty in applyingknowledge to slightly differentsituations.Difficulty in applyingknowledge to slightly differentsituations.Misconceptions remain.Random equationsSolving the “wrong” problemeither through oversimplificationor misreading ofthe problem.Frequent algebraic mistakes.Mathematical “magic” insolutions.Lack of interest or intellectualinvolvement.Difficulty visualizing physicalsituations.No pictures given.Situation realistic.Situation realistic.Reasonable motivation.Avoid “physics” words suchas inclined plane, inelasticcollision, frictionless, . . .Realistic situation thatreduces in a straightforwardway to a simple situation.Decisions necessary todetermine which concepts toapply and which quantitiesare relevant.Realistic situation described.Misconception will prevent acorrect solution.Problem requires more thanone mathematical andlogical step.The solution that does notobviously repeat the patternof textbook examples.Problem can be solved in astraightforward way usingfundamental physics.Actors are “you” andunnamed acquaintances.Situations are perceived asrealistic.There is a motivation.

Chapter 3: Combating Problem Solving That Avoids Physics 33For an expert, such as you, the two problems are identical and are solvedidentically. For the introductory student, however, they are very different. Thesecond form of the problem actually encourages students to use their novice,plug-and-chug or pattern-matching tactics. The student is not allowed topractice decision-making because the problem is already broken down intosingle-concept steps.This problem is difficult for students to visualize because no realistic context isgiven. Most importantly, a problem stripped down to its essentials does nothelp students connect physics with other parts of their knowledge. 2The first two columns of the table in Figure 3.3 give a summary of whatresearch indicates are some major difficulties students have solving physicsproblems. 3 The last column summarizes specific features of context-richproblems designed to help students overcome these difficulties.How to Discourage Students From Indiscriminate FormulaMemorization.Many students believe that solving a physics problem requires them to know theright equation(s) for that particular problem. So part of their plug-and-chug andpattern-matching techniques is the memorization of all the equations in eachchapter of their textbook. These equations are of equal importance to students.They do not distinguish the mathematical formulations of fundamentalprinciples from equations that are the consequence of those fundamentalprinciples to specific circumstances or for specific types of interactions. Worseyet, the equations memorized for one chapter are promptly forgotten whilememorizing the formulas for the next chapter.So how should the Third Law of Education be applied here?Make it easier for students to do what you want them todo and more difficult to do what you don’t want.What can you do to promote the development of a coherent, connected networkof knowledge organized around fundamental concepts, while discouraging thememorization of disconnected formulas?At first glance the answer may seem simple: allow students to write equations ona single file card or sheet of paper and bring their equations to your exams andexaminations. While this solution discourages memorization, it does not addressthe difficulty students have in developing a coherent knowledge base. Thosewho have tried this technique know too well that students will write as small aspossible to cram everything onto a single piece of paper. They then complain

34 Part 1:Teaching Physics Through Problem SolvingFigure 3.4. Example information sheet for exams: algebra-based courseThis is a closed book, closed notes quiz. Calculators are permitted. The only formulasthat may be used are those given below. Credit is given only for logical and completesolutions that are clearly communicated. In the context of a complete solution, partialcredit will be given for a well-communicated solution based on correct physics.5 points: A useful picture, defining the question, and giving your approach.7 points: A complete physics diagram defining the relevant quantities,identifying the target quantity, and specifying the relevant equations.6 points: Planning the solution by constructing specific equations and checking forsufficiency5 points: Executing the plan to get algebraic and numerical answer2 points: Evaluating the validity of the answer.Useful Mathematical Relationships:For a right triangle: sin = a c , cos = b c , tan = a b ,a 2 + b 2 = c 2, sin 2 + cos 2 = 1For a circle: C = 2πR , A = πR 2For a sphere:A = 4πR 2 , V = 4 3 πR3If Ax 2 + Bx + C = 0, then x = -B ±B2 - 4AC2AFundamental Concepts:rv r tvarr tvr = lim(∆t 0) ∆r∆tar = lim(∆t 0) ∆v r∆t F r ma rE system E transferE f E i E in E outKE 1 2 mv2Under Certain Conditions:vrvri v2a v 2rrfF k F NF s s F NF krF Gm 1m 2r 2F k eq 1 q 2r 2E transfer F r rPE mgyPE 1 2 kx2PE Gm 1m 2rPE k eq 1 q 2rUseful constants: 1 mile = 5280 ft, 1 ft = 0.305 m, g = 9.8 m/s 2 = 32 ft/s 2 ,1 lb = 4.45 N, G = 6.7 x 10 -11 N m2/kg 2 , k e = 9.0 x 10 9 N m 2 / C 2

Chapter 3: Combating Problem Solving That Avoids Physics 35that they would have done better on the exam if they had just had the foresightto write down one more equation.We have found it useful to supply students with only the information they needto solve exam problems from the basic ideas emphasized by the course. Anexample of such an information sheet used in an algebra-based course is shownin Figure 3.4. There are three features of the information sheet that aredesigned to promote student development of a coherent, integrated network ofknowledge and reinforce logical analysis of a problem using fundamentalconcepts.Supply a limited number of equations that students may use.Supply only the equations that state the fundamental physics principles andconcepts that are stressed in the course. Students are not allowed to use anyother equations to solve a problem. A distinction is made between physics thatunderlies everything and the important physics that depends on a specific butreasonably general situation. The symbols are not defined to encourage studentsto know the meaning of the equations.The choice of equations depends on the course and the emphasis of theinstructor. For example, the kinematics equations in Figure 3.4 would be givento the students in the algebra based physics course. In the calculus-basedcourse, we replace the equation for the special case of constant accelerationv vvxi xfx 2that builds on students intuitive understanding of an average withx12f axt voxt xo,2an equation directly connected by calculus to the solution of the equationdefining acceleration. [See Chapter 11, page 141 for an example of aninformation sheet for a calculus-based course.The information sheet grows with time.Nothing is ever taken off the sheet, but new equations and needed constants areadded. The “old” equations are often used in problems for new topics toemphasize that the underlying physics does not depend on the context. Forexample, the information that is not shaded was supplied for the second exam inthe course, the information in light-gray was added by the end of 8 weeks, andthe information in darker gray by the end of about 12 weeks in the course.Allow students to use calculators.Students love their calculators and feel

36 Part 1:Teaching Physics Through Problem Solvingpersecuted if they are not allowed to use them on exams. Unfortunately moderncalculators can hold an enormous amount of information. Since all studentshave an information sheet with all the equations that they are allowed to use,there is no advantage in using the calculator memory to store equations forexams. The same holds true for any temptation to bring written crib sheets toexams. If students are also required to communicate their solution in a logicaland complete manner, the ability and temptation to cheat is almost completelyeliminated. For these reasons, we see no benefit of restricting the use ofcalculators in CPS.Endnotes123The definition of a problem and problem solving has not changedsubstantially in the last forty years. For example, see Newell, A., & Simon,H.A. (1972), Human problem solving. Englewood Cliffs, NJ, Prentice-Hall, Inc;Hayes, J.R. (1989), The complete problem solver (2nd ed.), Hillsdale NJ; Lawrence;and Martinez, M. (1998), What is problem solving? Phi Delta Kappan, 79, 605-609.A number of researchers have concluded that the typical problems assignedin physics courses are actually counterproductive to learning physics. Othertypes of “nonspecific goal problems” and “ill-structured problems” havebeen found to move students towards more expert-like problem solving. See,for example, Sweller, J., Mawer, R. & Ward, M. (1982), Consequences ofhistory: Cued and means-end strategies in problem solving, American Journal ofPsychology, 95(3), 455-483; and Shekoyan, V., and Etkina, E. (2008),Introducing ill-structured problems in introductory physics recitations,Proceedings of the 2007 Physics Education Research Conference, PERC Publishing,Rochester, NY, 951, 192-195. In our own research, we also found thatstandard problems seemed to promote the use of the novice, plug-and-chugor pattern-matching strategies rather than the use of a more logical andorganized strategy. See Heller, P. & Hollabaugh, M. (1992), Teachingproblem solving through cooperative grouping. Part 2: Designing problemsand structuring groups, American Journal of Physics, 60(7), 637-644.For reviews of student difficulties solving problems, see Maloney, D.P. (1994),Research on problem solving in physics, in D.L. Gabel (Ed.), Handbook of Researchin Science Teaching and Learning, (pp. 327-354), NY, Macmillan; and Woods, D.(1989), Problem solving in practice, in D.L. Gabel (ed.), What Research Says to theScience Teacher Vol. 5, National Science Teachers Association. See also, Hsu, L.,Brewe, E., Foster, T. M., & Harper, K. A. (2004), Resource letter RPS-1:Research in problem solving. American Journal of Physics, 72(9), 1147-1156.

37Chapter 4Building Content intoProblem SolvingIn this chapter: An answer to the question: What is a problem-solving framework? An example of a problem-solving framework for introductory physics An example of an “ideal” student’s solution using this framework Problem solving as a series of translations Initial objections to teaching a problem-solving frameworkWhether or not a question is aproblem depends on theviewpoint of the person seeking asolution. If that person knows how toarrive at a solution, even if they don’t knowthe answer, then the question is not aproblem for them .1 If the question is not aproblem, it is not necessary to use a logicalproblem-solving framework thatemphasizes the understanding offundamental concepts and the practice ofimportant problem solving skills. This isthe case whether or not a person’s solution is correct. It takes a great deal ofconfidence for a person to approach a question as a problem and embark on asolution based only on physics, logic, and mathematics without knowing the paththat will be travelled. Most people require the additional guidance of a frameworkto embark on this journey.Unfortunately most students do not come into ourintroductory physic classes with a robust andlogical general problem-solving framework. A sureindication is when students tell you they don’t evenknow how to get started on a problem. Without areasonable framework, students little choice otherthan to continue to apply the unproductive novicestrategies with which they entered the course. [SeeChapter 2 for a description of the plug-and-chugand pattern-matching novice strategies for solvingproblems.]When is a questiona “problem?”If you know how toarrive at a solution,even if you don’t knowthe answer, then thequestion is NOT aproblem for you.

38 Part 1:Teaching Physics Through Problem SolvingConsequently, if your goal is to teach physics through problem solving, thenyour students will need an explicit example of a problem-solving framework thatdirects their efforts toward making connections both among physics conceptsand between those concepts and the rest of their knowledge. The frameworkshould be a logical and organized guide to build a problem solution. It getsstudents started, guides them to what to consider next, organizes theirmathematics, and helps them determine if their answer is correct.In this chapter, we first describe a general problem-solving framework used byexperts in all fields, then give an example of that framework tailored to ourintroductory, algebra-based physics course. If you want to explore what it is liketo use a problem-solving framework, the next section provides an example of an“ideal:” student’s problem solution arrived at by using our framework. Includedin the example is a commentary explaining the purpose and rationale for eachpart of the framework. In Chapter 14 we describe how you could personalize aproblem-solving framework for the introductory physics course you teach.What Is a Problem-solving Framework?There is no formula for true problem solving.Problem solving is a process similar to working yourway through an unfamiliar forest. You navigate yourway step by step, making some false moves but graduallymoving closer toward the goal. Each step is morelikely to succeed the choice is guided by somefundamental principle. But what are these “steps” andwhat guides your decisions?We have known for a long time that humans generallyfollow the same steps to solve any problem. Manypsychologists and educators have described these stepsin slightly different ways. 2 One of the most influentialdescriptions is by the mathematician George Polya (1945): 31. Understand the Problem (i.e., define the problem)2. Devise a Plan3. Carry Out the Plan3. Look Back (i.e., check your results)Of course, these steps are a simplification of a complex process. A personsolving a problem overlaps steps. For example, you may begin to devise a planwhile you are defining the problem. You also backtrack to earlier steps. Forexample, in devising a plan for a solution, you may decide that you have not

Chapter 4: Building Physics into Problem Solving 39Figure 4.1. The general problem-solving framework used by experts in all disciplines.Step 1. Understand the ProblemBring the problem into focus by describing the situation and goal(s) asprecisely as possible.Describe the problem in the terms developed by your field of expertise. Translate the situation and goals into the fundamental concepts of yourfield using the notation developed by your field. Decide the reasonable idealizations and approximations you need tomake.Step 2. Devise a PlanApply the specialized techniques (heuristics) of your field to develop aplan, using the concepts of your field to connect the situation with thegoal.Re-examine the description of the problem if a solution does not appearpossible.Step 3. Carry Out the PlanFollow your plan to the desired result.Re-examine your plan if you cannot obtain the desired result.Step 4. Look Back.Determine how well your result agrees with your knowledge ofsimilar behavior, use limiting behavior that you understand.fully understood the problem and go back to consider the situation. When youcarry out your plan, you may find that the plan is not complete or that it reallydoes not lead to a solution, so you modify the plan. Finally, you may skip somesteps altogether, depending on your background and experience.You probably recognize Polya’s steps as self-evident. But what guides ourdecisions through the several possible paths in our problem-solving? Polyaintroduced the word heuristics for the thinking tools by which problems aresolved. A heuristic is a rule of thumb – a strategy that is both powerful andgeneral, but not absolutely guaranteed to work.For example, one general heuristic used in Step 1 (understanding the problem) isto determine the goal, the unknowns, the data (givens), and the conditions thatrelate the data. Another heuristic used to devise a plan (Step 2) is called workingbackwards.1 Start with the ultimate goal and then decide what would constitutea reasonable step just prior to reaching that goal. Then ask yourself what thestep would be just prior to that, and so on until you reach the initial conditionsof the problem. Another general heuristic is to break a problem into subproblemsthat you can solve.

40 Part 1:Teaching Physics Through Problem SolvingIf you are thinking that you do not use generalheuristics to solve textbook problems, then youare right. For you, textbook problems, are notreal problems. You know roughly what path totake to get a solution. 4 You can work forwardsfrom fundamental physics concepts and theknown information towards the solution. 5When faced with an atypical or novel situation,however, it is a problem for you. You thenprobably use techniques similar to the Polya’sgeneral heuristics. It is often difficult for expertproblem solvers to articulate these techniquesbecause they are often automated and deeplyintegrated into a large and specialized knowledge structure. For example, expertphysicists faced with an unfamiliar or novel problem will often: 6 Use analogies with systems they understand better. Search for potential limitations to the analogy. Refer to mental models based on visual and kinesthetic “intuition” to try tounderstand how the target system would behave. Investigate the target system with extreme-case arguments, probing howthe system would work if different parameters were pushed to zero orinfinity.And, of course, when faced with unfamiliar or novel problems, at some pointexperts work backwards from the target unknown, dividing the problem intosub-problems that can be solved. It turns out that experts in all fields solveunfamiliar or novel problems in a similar way. This general problem-solvingframework is shown in Figure 4.1.A Physics Example: The Competent Problem-solving FrameworkBecause you are a expert problem solver, you probably do not pay muchattention to your own problem-solving framework, outlined in general terms inFigure 4.1. You don’t even have to use a problem-solving framework to solveintroductory physics problems because they are not really problems for you.However, if you want your students to solve problems as a tool for learningphysics, they will need to use a problem-solving framework that emphasizes theapplication of fundamental concepts and the connection of those concepts totheir existing knowledge using generally useful problem solving skills.

Chapter 4: Building Physics into Problem Solving 41Figure 4.2. The Competent Problem-solving Framework: Algebra Version1. Focus the Problem. Establish a clear mental image of the problem.A. Visualize the situation and events by sketching a useful picture.• Draw a picture showing how the objects are related spatially, they are moving, andthey are interacting. The drawing should show the time sequence of events,especially for those times when an object experiences an abrupt change.• Write down the relevant known and unknown information, giving each quantity asymbolic name and adding that information to the picture.B. Precisely state the question to be answered in terms you can calculate.C. Identify physics approach(es) that might be useful to reach a solution.• Which fundamental principle(s) of physics (e.g., kinematics, Newton’s Laws,conservation of energy) might be useful in this situation.• List any approximations or problem constraints that are apply to this situation.2. Describe the PhysicsA. Draw any necessary diagrams with coordinate systems that are consistent with theapproach(es) you have chosen.• Define consistent and unique symbols for any quantities that are relevant to thesituation.\B. Identify the target quantity(s) that will provide the answer to the question.C. Assemble the appropriate equations to quantify the physics principles and constraintsidentified in your approach.3. Plan a SolutionA. Construct a logical chain of equations from those identified in the previous step,leading from the target quantity to quantities that are known.• Begin with the quantitative relationship that contains the target variable. Identifyother unknowns in the equation.• Choose a new equation for one of these unknowns. Keep track of any additionalunknowns.• Continue this process for each unknown.B. Determine if this chain of equations is sufficient to solve for the target quantity bycomparing the number of unknown quantities to the number of equations.C. Write down a verbal description of the solution steps you will take to solve this chain ofequations so that no algebraic loops are created. Work from the last equation to thefirst equation that contains the target quantity.4. Execute the PlanA. Follow the outline from in the previous step.• Arrive at an algebraic equation for your target quantity by following your verbaldescription of the solution steps.• Check the units of your final algebraic equation before putting in numbers.• If quantities have numerical values, substitute them in your final equation tocalculate a value for the target quantity.5. Evaluate the AnswerA. Does the mathematical result answer the question with appropriate units?B. Is the result unreasonable?B. Is the answer complete?This framework should be designed so that students practice an expert problem

42 Part 1:Teaching Physics Through Problem Solvingsolving behavior of examining the conceptual aspects of the problem beforelaunching into mathematical calculation. This causes each student to examinetheir physics knowledge to which helps remediate weakly held misconceptionsand prevents new ones from forming. Such a framework should also emphasizethat mathematical calculation is only one small part of a problem solution.Introductory students do not tend to generalize new techniques, so theframework that you want them to use must be explicitly demonstrated each timeyou introduce a new topic. Constructing such a problem-solving frameworkmay be difficult for you because you do not need a framework to solveintroductory physics questions because they are not problems for you.Fortunately, there are several available implementations of the Polya’s generalproblem-solving framework that have been used to help teach introductoryphysics. You can modify one of these to fit your taste and the needs of yourstudents.Several classroom studies show that the explicitteaching of these frameworks does result inbetter problem solving. 7 , 8 A concrete exampleof a specific implementation of a problemsolving framework for students in our algebrabasedcourse is shown in Figure 4.2. We basedthis framework, called the Competent ProblemsolvingFramework, on our own research and theresearch of others on student difficulties in solving physics problems. [SeeChapter 14, page 173 for the problem-solving framework we use in our calculusbasedcourse for scientists and engineers.] In particular this framework isconstructed to address specific weaknesses in the problem solving of thosestudents based on the analysis of written problem solutions, informal interviewsof students during office hours, and observations of students working incooperative groups. Specific parts of the framework were tested by removingthem to see if the student difficulties reoccurred.The Competent Problem-solving Framework has five steps. Each step consistsof specific actions that lead the student to decisions that confront their difficultiesand guide them to the next decision point in the solution. You will, no doubt,recognize most of the actions as things you expect your students to do. Theplacement of the decisions in one step rather than another is an artificialconsequence of having to delineate steps. For a true expert, this process is acontinuous whole.Our problem-solving framework, like all other research-based frameworks, ismore proscriptive than the framework you use to solve unfamiliar problemsbecause it is designed both as a tool for learning physics and to move studentsfrom novice problem solving toward a more expert-like problem solving. 9 Thisis why we call it the Competent Problem-solving Framework. 10 We do not expectstudents to become expert problem solvers by the end of an introductoryphysics course.

Chapter 4: Building Physics into Problem Solving 43The details of the framework that you decide to teachshould be tailored to the needs and backgrounds of yourstudents and to your own approach to your introductorycourse. Chapter 14 provides some suggestions abouthow you could personalize a problem-solvingframework.

44 Part 1:Teaching Physics Through Problem SolvingExample of an “Ideal” Student’s Problem Solution for an AlgebrabasedCourseTo be more concrete, Figure 4.3 (pages 42 – 49) shows how to use theCompetent” Problem-solving Framework to solve a problem suitable forintroductory students at the beginning of an algebra-based physics class. Thesolution is on the left (even) side of the page. The right (odd) side of the pagecontains commentary about how the details would support particular goals in anintroductory course. [See Chapter 14, pages 175 for a problem-solvingframework for the calculus-based course.]As you read this example, you may want to take some notes to help youpersonalize a problem-solving framework for your own students and situation,as described in Chapter 14. What parts of this Competent Problem-solvingFramework matched the goals for your introductory course? What parts did notmatch your goals? What parts are not needed by your students? What ismissing that your students need? What wording needs changing to better matchyour students? If so, how would you change it?

Chapter 4: Building Physics into Problem Solving 45Figure 4.3. Example of an “ideal” student problem solution, with commentaryThe Problem: You are driving at 50 mph on a freeway when you wonder what your stoppingdistance would be if the car in front of you jammed on its brakes. When you get home youdecide to do the calculation. You measure your reaction time to be 0.8 seconds from the timeyou see the car’s brake lights until you apply your own brakes. Your owner’s manual says thatyour car slows down at a rate of 6 m/s2 when the brakes are applied.Step 1: FOCUS ON THE PROBLEMDraw a picture, identifying the useful quantities.seeslight0.8 secappliesbrakesstoppedv o = 50 mph a = 6m/s2 v f= 0stopping distanceQuestion:What distance did the car travel from when thebrake light is seen to when it stopped?Approach: Use the definitions of velocity and acceleration.The velocity is constant until the brakes are applied. Inthis time interval, the average velocity is equal to theinstantaneous velocity.The acceleration is constant after the brakes are applied.In this time interval, the average velocity is not equal tothe instantaneous velocity. However, the averageacceleration is equal to the instantaneous acceleration.

46 Part 1:Teaching Physics Through Problem SolvingFigure 4.3 (continued). Example of an “ideal” student problem solution, with commentaryCommentaryThe Problem. This problem emphasizes the basic definitions of velocity andacceleration in one dimension. It also emphasizes the difference betweeninstantaneous kinematics quantities and their average values, a concept is verydifficult for students. The problem gives the students practice in contrastingconstant velocity motion with that at a constant acceleration. Students also needpractice using the units of physical quantities. We give students problems withmixed sets of units so they will pay attention to them. Every specialty uses itsown set of units for either historical reasons or convenience. From ourquestionnaire, we found that the faculties from departments that require theirstudents to take our physics course want the physics course to show theirstudents how to convert and manipulate units.Picture. Students’ difficulty with visualization can be immediately seen bytheir difficulty in drawing a useful picture. At the beginning of the course, manystudents need to draw very realistic looking objects such as cars while others aresatisfied with more expert-like drawings using a simple symbol, such as arectangle, to represent a car. Almost all beginning students have difficultydrawing multiple images on a single picture to represent an object at multiplepositions of interest. They have difficulty making a decision, as to where thoseinteresting positions might be. Many can verbalize these features before theycan indicate them on a drawing. Students also have difficulty associating theirpictorial representation with quantities that represent the object’s motion. Weemphasize to our students that once they have drawn the picture, then theyshould not have to read the problem again. This makes for better timeefficiency in solving the problem.Question. Writing the question in their own words helps prevent studentsfrom solving for some quantity that is not desired. Here the student mustdecide on a reasonable definition of “stopping distance.”Approach. Writing an approach helps students concentrate on the importantphysics in the problem.

Chapter 4: Building Physics into Problem Solving 47Figure 4.3 (continued). Example of an “ideal” student problem solution, with commentaryStep 2: DESCRIBE THE PHYSICSMake a diagram of the situation, defining the quantities thatphysics uses to describe motion (velocity and acceleration ateach interesting position and time on a coordinate system).a = 6m/s2a = 6m/s2v o = 50 mphx o = 0t o = 0v 1 = 50 mph v 2 = 0x 1 = ?t 1 = 0.8 s+ xx 2 = ?t 2 = ?Target quantity: x 2Possibly useful equations:v x xt, for constant velocity v x v xfor constant acceleration v x v i v f2a x v xt, for constant acceleration a x a x

48 Part 1:Teaching Physics Through Problem SolvingFigure 4.3 (continued). Example of an “ideal” student problem solution, with commentaryCommentaryMake a diagram of the situation. Students have a great deal ofdifficulty associating an object with a specific acceleration and velocity at aspecific position and time. This diagram helps them because it requires thedrawing of an idealized object, now a point, at each interesting position. Thediagram also begins the process of getting students to define an appropriatecoordinate system for a situation. It gives the students practice in the process ofgoing from real objects to idealized objects. For experts there is no significantdifference between the picture and the diagram, but for beginning students thereis. The picture gives practice in visualizing the behavior of real objects. Thediagram, on the other hand, gives students practice with the visual relationshipsof physical quantities. As time goes on in the course, more and more studentsbecome more expert-like and combine these two types of images. At thebeginning of the course, most of our students are not ready to combine thesetwo types of visualization. Those who try to combine the picture and thediagram tend to become more and more confused as situations become morecomplex.Target quantity. Writing down this quantity gives students a focus for theirmathematics. When doing mathematical manipulations many students lose trackof the quantity they want.Possibly useful equations. This is where the student explicitly connectsthe conceptual approach to the problem with mathematics. It represents agathering of mental resources or a “toolbox” for the quantitative solution to theproblem. Even in the beginning chapter on kinematics in most textbooks, thenumber of equations students believe they will need to know can overwhelmthem. Here they must decide to limit those equations to only those that areindependent and might be useful. At this stage, it is OK if there are some extraequations since, the student does not yet know how to solve the problem.To compel students to concentrate their efforts on the basic concepts anddiscourage the student behavior of formula memorization, we only allow ourstudents to use equations chosen by the instructor (see Chapter 3, pages 31 -33). The choice of equations depends on the course and the emphasis of theinstructor. For example, the equations used in this solution would be given tothe students in the algebra based physics course. In the calculus-based course,we replace the equationv vvxi xf1 2x with xf axt voxt xo,22an equation directly connected to the calculus expression defining acceleration.

Chapter 4: Building Physics into Problem Solving 49Figure 4.3 (continued). Example of an “ideal” student problem solution, with commentaryStep 3: PLAN THE SOLUTIONConstruct the chain of equations giving a solution. Beginwith an equation containing the target quantity. Keep track ofany additional unknown quantities that are introduced.UnknownsFind x 2x 2v 1, 2 x 2 x 1v1,2 , x 1, t 2t 2 t 1Find v 1,2v 1,2 v 1 v 22 v 12Find x 1v 1 x 1 x ot 1 t o x 1t 1Find t 2a 1 v 2 v 1t 2 t 1 v 1t 2 t 1Check for sufficiency:Yes: 4 unknowns and 4 equationsOutline the solution steps. Work from the last equation to thefirst equation that contains the target quantity.Solve for t 2and put it into .Solve for x 1and put it into .Solve for v 1,2 and put it into .Solve for x 2.

50 Part 1:Teaching Physics Through Problem SolvingFigure 4.3 (continued). Example of an “ideal” student problem solution, with commentaryCommentaryConstruct the chain of equations giving a solution. We wish toconvince students that solving a problem relies more on understanding theconcepts of physics than on mathematical techniques. Students seem to trustmathematical manipulation to give them an answer and make “magical” mistakesto get one. This procedure restrains that manipulation by requiring thatsufficient equations to solve the problem have been assembled first. It does notalways yield the most elegant solution, but it is straightforward, easy tounderstand, and very general.Students always know where to begin because they always start with an equationthat contains the target quantity. The unknowns in that equation give thestudent a way to decide on next equation to be used and so on. This proceduregives students a logical way to build a chain of equations that connects the targetquantity to quantities that are known. If the process reaches a “dead-end”, thestudent explicitly see how to revise one of their decisions of which equation isused to determine an unknown quantity. This procedure depends on havinglinearly independent equations, which is another reason for the instructor tocontrol the equations that can be used (see Chapter 3, pages 31 - 33). Althoughwe emphasize paper and pencil solutions, constructing this chain of equations isvery useful for computer or calculator algebra.Remember that working backwards from the goal is a general thinking tool(heuristic) used by experts solving real problems. See the reference in Endnote1 for a description of working backwards and other heuristics.Check for sufficiency. Matching the number of equations with thenumber of unknowns gives the students an easy way to determine if they needmore information to solve the problem. After a few weeks, we also point outhow you can solve the problem if there are fewer equations than unknownsprovided an unknown cancels out. We give instruction on how to detect suchcases and the physics implication of these cases.Outline the solution steps. Actually writing down an outline seems tobe necessary for most of our algebra-based physics students, but not for thecalculus-based students. Students who do not trust their mathematicsbackground can become very confused when doing algebra unless they have awritten plan. All introductory students often get into infinite algebraic loopswhen solving equations with more than two unknowns. Here when an unknownis determined, even in terms of other unknowns, it is immediately substituted inevery upstream occurrence. This procedure assures that such algebraic traps areavoided.

Chapter 4: Building Physics into Problem Solving 51Figure 4.3 (continued). Example of an “ideal” student problem solution, with commentaryStep 4: EXECUTE THE PLANFollow the outline from Step 3.Solve for t 2: t 2 t 1 a 1 v 1t 2 t 1 v 1a 1t 2 v 1a 1 t 1Solve for x 1: v 1 x 1t 1v 1 t 1 x 1Solve for v 1,2 : v 1,2 v 12Put into and solve for x 2: v 1,2 x 2 x 1t 2 t 1v 12 x 2 v 1 t 1v 1 t 1 t 1a 1v 1v 1 x2 v2 a 1 1 t 1v 1 t 1 v 1 2 x2a 21Calculate the value of the target quantity.x 2 50 mi 50 mi 2 hr 0.8s hr 2 6 m s 2 x 2 50 mi50 mi2 1609mhr min 1609mhr min hr mi 60 min60s 0.8s hr mi 60 min60s 2 6 m s 2 x 2 18m 42m 60mStep 5: EVALUATE THE ANSWERx 2is the distance traveled by the car from when brake light isseen to stopping. The question is answered.The answer is in meters, a correct unit of distance.A car is about 6 meters long so 10 car lengths is not anunreasonable distance to stop a car going that fast.

52 Part 1:Teaching Physics Through Problem SolvingFigure 4.3 (continued). Example of an “ideal” student problem solution, with commentaryCommentaryStep 4 - Execute the Plan.This is the only part of the problem containing mathematical manipulation. Themathematical solution follows the verbal outline. Here the student begins withquantities that are known and proceeds backwards through the chain ofequations to the target quantity. The student can concentrate on themathematics because they are assured of a solution when they follow the plan.Step 5 - Evaluate the AnswerThis step reinforces the connection of the physics used in the problem solutionto the student’s reality. It is the step most characteristic of expert problemsolving and is the most difficult part of problem solving for introductorystudents.

Chapter 4: Building Physics into Problem Solving 53Problem Solving as A Series of TranslationsIt is often helpful to have more than one way to think about something. Anyproblem-solving framework, including the Competent Problem-solvingFramework, can be viewed as a series of actions to help students makeincreasingly abstract mental translations (steps) from the situation to an answer.In our framework, the first actions in each step define the mental translation.The last action in each step is a “bridge” to prepare students for the translationin the next step. The series of mental translations is outlined below.Step 1. Focus on the ProblemTranslate from a situation to an image of the situation, including importantinformation given in the problem.Prepare for the next step by identifying theapproach to the problem.Step 2. Describe the PhysicsTranslate from the image of the situation to a diagrammatic representation of thesituation.Define symbols of the quantities used by physics to describe a situation.

54 Part 1:Teaching Physics Through Problem SolvingPrepare for the next step by identifying the equations useful in describing thesituation.Step 3. Plan a SolutionTranslate from a physics representation to a mathematical representation of thesituationPrepare for the next step byidentifying the mathematical stepsnecessary to reach a solution.Step 4. Execute Your PlanTranslate from a mathematical representation of the situation to a set ofmathematical actions that yield a solution.

Chapter 4: Building Physics into Problem Solving 55Prepare for the next step by checking unitsduring the mathematical process.Step 5. Evaluate Your SolutionTranslate from a mathematical solution to an answer connected to otherknowledge.Initial Objections to Teaching a Problem-solving FrameworkAs you were reading through the example of the Competent Problem-solvingFramework, you may have had the following thoughts:1. If students use this framework, they will write a lot in their problemsolution. Won’t this make them more difficult and time consuming tograde?Actually our most common feedback from instructors using a framework similarto the example is that because students write more, student solutions aremuch easier and less time consuming to grade. This is because the solutionsare more organized and easier to follow. Since student thinking is nowexplicit, it becomes more straightforward to determine where a student hasgone wrong and grade accordingly. This also makes grading more useful tothe instructor and allows that instructor to give more useful feedback to thestudent.2. This framework is too long and complex. My students will see it as justadding busy work that slows their ability to solve a problem. They won’tuse it.There is no getting around this. Expert thinking is a long and complex processwhen analyzed. However, with repetition, the mind combines many of thesubprocesses together and automates them so that they happen very rapidly.To build toward expert-like problem solving, students need to practice thesesubprocesses. Because this is a fundamental change of their ideas aboutproblem solving, students must begin by implementing each part of expertlikeproblem solving in a conscious manner. This will overload their shortterm memory which requires them to write down their thinking and, moreimportantly, use what they have written to solve the problem. As withlearning any new skill, this does slow them down. This can be frustratingfor students and cooperative groups [see Chapt. Xxx] and continuousmodeling of the process provide important support to get over this hurdle.Working in cooperative groups, students quickly find that they can make

56 Part 1:Teaching Physics Through Problem Solvingprogress toward solving problems that were previously well beyond theircapability. After practicing such a framework for a month or two, both ingroups and alone, their individual speed increases to that of their noviceattempts at a solution with a much higher probability of success. In a singleyear, students will not achieve the mental automation of an expert but canachieve a level of competent problem solving. The key to making progressrenovating mental structure is practicing that structure. As Vince Lombardisaid, “Practice does not make perfect, only perfect practice makes perfect.”3. This framework reduces problem solving to following a recipe. It takescreative thinking out of problem solving.The framework only points out the places in a problem solution where decisionsneed to be made. Since each decision can lead to a different path to asolution, one student’s solution can be very different from another’s. Byexplicitly showing the decision points, the framework emphasizes then needfor creative thought in problem solving.All research based problem-solving frameworks in physics are specificimplementations of the Polya’s general framework (Figure 4.1). They were,however, developed and tested for different populations of students. They dividethe important actions into a different number of steps and sub-steps, describethe same actions in different ways, and emphasize different heuristics dependingon the background and experiences of the intended population of students. Thedetails of the framework that you decide to teach should be tailored to the needsand backgrounds of your students and to your own approach to your introductorycourse. Chapter 14 provides some suggestions about how you could personalizea problem-solving framework, including some examples of different frameworksin physics textbooks.Remember that breaking any activity down into its small but necessary steps, likestarting a car in the winter, makes it seem very complex (see Figure 5.1, page56). At first your personalized problem-solving framework will certainly seemso to your students. For students to use any such framework, the problemsmust be constructed so that novice strategies fail and support, such ascooperative groups, must be available to allow students to be successful fromthe beginning. This book contains suggestions for how to change the structureof your course, including Cooperative Problem Solving, so your students will useyour personalized problem-solving framework.We have already made two suggestions in Chapter 3, design context-richproblems for your students to solve, and limit the given equations students canuse on an exam. In the next chapter we describe several course changes thatmake it easier for your students to use your personalized problem-solvingframework rather than their own novice strategies. The chapters in Parts 2 and3 describe how to structure and manage Cooperative Problem Solving.

Chapter 4: Building Physics into Problem Solving 57Endnotes1 Martinez, M. (1998). What is problem solving? Phi Delta Kappan 79, 605-609.2 Some of the well known descriptions of the stages or “steps” in human problemsolving are by: Wallis, G. (1926), The art of thought, NY, Harcourt, Brace &World; Dewey, J. (1933), How we think, NY, Heath; Kingsley H.L. & Garry R.(1957), The nature and conditions of learning, 2 nd ed., NJ, Prentice-Hall; andBransford, J.D. & Stein, B.S. (1984), The ideal problem solver: A guide for improvingthinking, learning, and creativity, NY, W. H. Freeman and Company.3 Polya, G. (1957). How to solve it: A new aspect of mathematical method, Princeton,NJ: Princeton University Press.4 For an excellent discussion of the special knowledge used by experts, seeBereiter, C. & M. Scardamalia, M. (1993), Surpassing ourselves: An inquiry into thenature and implications of expertise, IL, Open Court, pages 25 –75.5 For reviews of expert-novice research, see: Maloney, D.P. (1994), Research onproblem solving in physics, in D.L. Gabel (ed.), Handbook of research in scienceteaching andlearning, (pp. 327-354), New York: Macmillan; and Hsu, L., Brewe, E.,Foster, T. M., & Harper, K. A. (2004), Resource letter RPS-1: Research inproblem solving. American Journal of Physics, 72(9), 1147-1156.6 See Clement, J. (1990), Non-formal reasoning in physics. The use of analogiesand extreme cases, in J. Voss, D. N. Perkins, and J. Segal (Eds.), Informal reasoning,NJ, Erlbaum; and Johnson, P.E., Ahlgren, A., Blount, J.P., & Petit, N.J. (1980),Scientific reasoning: Garden paths and blind alleys, in J. Robinson (Ed.), Researchin Science Education: New questions, new directions, Colorado Springs, CO, BiologicalSciences Curriculum Study.7 See, for example, Mettes, C.T.C.W., Pilot, A., & Roossink, H.J. (1981),Linking factual and procedural knowledge in solving science problems: A casestudy in a thermodynamics course, Instructional Science, 10, 333–361; vanWeeren, J.H.P., de Mul, F.F.M., Peters, M.J., Kramers-Pals H., & Roossink,H.J. (1982), Teaching problem-solving in physics: A course inelectromagnetism," American Journal of Physics, 50, 725–732; Heller, J.I. & Reif,F. (1984), Prescribing effective human problem-solving processes: Problemdescription in physics, Cognitive Instruction, 1, 177–21: Bascones, J., &Novak,J.D. (1985), Alternative instructional systems and the development ofproblem-solving skills in physics, European Journal of Science Education, 7, 253–261; Wright, D.S., & Williams, C.D. (1986), A WISE strategy for introductoryphysics, Physics Teacher, 24, 211–216; Van Heuvelen, A. (1991), Overview,case study physics, American Journal of Physics, 59(10), 898-907; Reif, F. (1995),Millikan Lecture 1994: Understanding and teaching important scientificthought processes, American Journal of Physics, 63, 17–32

58 Part 1:Teaching Physics Through Problem Solving8 Research indicates that curriculum innovations that incorporate one or morecommon features result in better problem-solving performance in students.These features are: (1) explicit teaching of problem-solving heuristics (similar tothe Competent Problem Solving Framework); (2) modeling the use of theheuristics by the instructor, and (3) requiring students to use the heuristicsexplicitly when solving problems. See Hsu, L., Brewe, E., Foster, T. M., &Harper, K. A. (2004). Resource letter RPS-1: Research in problem solving.American Journal of Physics, 72(9), 1147-1156, Section IV.9 For the stages between novice and expert, see Endnote 4.10 Heller, K. & Heller, P. (2000). The competent problem solver for introductory physics:Algebra, McGraw-Hill Higher Education.

59Chapter 5Reinforcing Student Use ofa Problem-solving FrameworkIn this chapter: Flow charts of problem-solving decisions. Answer sheets with cues from the problem-solving framework Example problem solutions worked out on the answer sheetsIn Chapter 4 we described a general problem-solving framework used by expertsin all fields, then gave an example of a problem-solving framework that we usein our introductory, algebra-based physics courses. A problem-solvingframework is a logical and organized guide to the decisions needed to build aproblem solution. It gets students started, guides them to what to consider next,organizes their mathematics, and helps them decide how to evaluate their answer.[See Chapter 14 for how to personalize a problem-solving framework for yourstudents.]Problem solving requires making many decisions. Breaking any activity down intothe actions necessary to accomplish it, makes it seem very complex (see Figure 5.1).At first a problem-solving framework will seem too complex to your students. TheThird Law of Education states: Make it easier for students to do what you want them to doand more difficult to do what you don’t want. How can you make a problem-solvingframework easier for your students to use?One answer is to present and reinforce your problem-solving framework in a varietyof ways. This helps students with different backgrounds and experiences build theirown mental models of the framework. In Chapter 4 we showed one way to presenta problem-solving framework. In this chapter we describe three additional ways todescribe and present your problem-solving framework to help your students becomemore comfortable using it. In the next chapter (Chapter 6)Figure 5.1. Steps for Starting a Car In Winter

60 Part 1: Teaching Physics Through Problem Solving

Chapter 5: Reinforcing Student Use of a Problem-solving Framework 61we explain why these techniques, while necessary, are not sufficient for students toadopt a logical, organized problem solving framework. This is the rationale forCooperative Problem Solving (CPS).Flow Charts of Problem-solving DecisionsWe commented earlier that, at first, a problem-solving framework seems verycomplex to your students. You need to reassure them that, once they get the hangof it, the framework is simpler and more natural than it appears. You might evenshow your students how a simple action, such as starting a car, looks complicated ifyou break it into steps. This example is shown in Figure 5.1. You might alsoremind them of the first time they drove a car. That it seemed like they had tocheck so many things and make so many decisions. Now they still have to check thesame things and make the same decisions but it seems effortless.Remember the 1st Law of Instruction: Doing something once is not enough.Demonstrating building a complete problem solution using a framework in yourlectures at the beginning of the course is not enough. You must demonstrate theframework every time you give an example problem solution. This will be at leastevery time you introduce a new topic. Students also need to practice using theframework to solve appropriate problems with scaffolding for beginners. Scaffoldingis structure that supports the learning of problem-solving skills. For example, wefound that the one-page outline of the Competent problem-solving framework(Chapter 4, page 37) was not enough information for the majority of our students tounderstand and begin to use the framework by themselves. Many students neededto read an explanation of each step and sub-step in the framework.We also found that these written explanations were also not enough for manystudents. We present each step of our Competent Problem-solving Framework as aflow chart of the actions in each step and sub-step, as illustrated in Figures 5.2athrough 5.2e. Next to each action are some decisions, in question form, thatstudents may need to make to complete each action successfully.Notice that the flow charts are very simple, and limitedto four or five actions. The last three steps includeonly one decision loop rather than the actual numberof loops. This is intentional. Our research indicatedthat students in introductory physics courses find flowcharts with more than one decision loop overwhelmingand they reject them as too complex.

62 Part 1: Teaching Physics Through Problem SolvingProblem-solving Answer SheetsAnother way to reinforce the use of a problem-solving framework is to provideworksheets for student solutions that include cues for the major steps of yourproblem-solving framework. Worksheets are another form of scaffolding or providestudents with another “ladder” (scaffolding) up the learning mountain.Figure 5.2a. Flow chart of Step 1 -- Focus on the Problem.

Chapter 5: Reinforcing Student Use of a Problem-solving Framework 63Problem StatementFocus onThe ProblemConstruct a mental imageof the sequences ofevents described in theproblem statement.• What's going on?• What objects are involved?• What are they doing?Sketch a picture thatrepresents this mentalimage; include giveninformation.• Are all the important objects shown?• Are the spatial relations between theobjects shown?• Are the important times represented?• Are the important motionsrepresented?• Are the important interactionsrepresented?Determine the question.• Does the question ask about a specificmeasurable characteristic(s) about aparticular object(s)? If not,reformulate it so it does.Select approach(es) thatyou think will lead to asolution of the problem.• What is the system of interest?• Which fundamental physics conceptscould be used to solve the problem?• What information is really needed?• Are there only certain time intervalsduring which one approach is useful?• Should you make any approximations?Describe the PhysicsFigure 5.2b. Flow chart of Step 2 – Describe the Physics

64 Part 1: Teaching Physics Through Problem SolvingFocus on the ProblemConstruct diagram(s) toshow important spaceand time relationships ofeach object.Make sure all symbolsrepresenting quantitiesshown on diagram(s) aredefined.Describethe Physics• What coordinate axes are useful? Whichdirection should be positive?• Relative to the coordinate axes, whereis (are) the object(s) for each importanttime?• Are other diagrams necessary torepresent the interactions of eachobject or the time evolution of its state?• What quantities are needed to definethe problem mathematically using theapproach chosen?• Which symbols represent knownquantities?Which symbols represent unknownquantities?• Are all quantities having differentvalues labeled with unique symbols?• Do the diagrams have all of theessential information from the sketch?Declare a target quantity.• Which of the unknowns defined on thediagram(s) answers the question?State mathematicalrelationships fromfundamental conceptsand specific constraints.• What equations represent thefundamental concept(s) specified in ourapproach and relate the physicsquantities defined in the diagram?• During what time intervals are thoserelationships either true or useful?• Are there any equations that representspecial conditions that are true forsome quantities in this problem?Plan a SolutionFigure 5.2c. Flow chart of Step 3 – Plan a Solution

Chapter 5: Reinforcing Student Use of a Problem-solving Framework 65Describe the PhysicsChose one of thequantitative relationshipsthat involves the targetvariable.Plan aSolution• Which quantitative relationshipincludes the target quantity?• For what object does that equationapply?• For what time interval does thatequation apply?Substitute specificvariable symbols intogeneral equations; dropvariables with zero value.NoAre thereadditionalunknowns?yesChoose a new equationthat involves the newunknown; substitutespecific variable symbols.• Are there any unknowns in theequation other than the targetquantity?• Are there any unknowns thatcancel out in the algebra?• Which quantitative relationshipincludes the unknown quantity?• For what object does that equationapply?• For what time interval does thatequation apply?• Is this equation different fromthose already used in this solution?Outline how to use thespecific equations todetermine the targetvariable.Execute the Plan• • Are there the same number ofequations as there are unknownquantities? If not, will one of theunknowns cancel out in thealgebra?• In what order should the equationsbe combined in order to solve forthe target quantity in terms of onlyknown quantities?Figure 5.2d. Flow chart of Step 4 – Execute the Plan

66 Part 1: Teaching Physics Through Problem SolvingPlan a SolutionExecute the PlanSelect the last (unused)equation from your plan,isolate the unknownquantity.• Start with the first equation inyour plan.• The last equation used in this loopshould contain the target quantityas the only unknown.Substitute thisrelationship into each ofthe other equations in theplan.• What unknown is the target ofthis specific equation?• Which the other unusedequations in your plan havethat unknown?• Are there any quantities thatcancel out in the algebra?NoHas the targetvariable beenisolated?yes• Has each of the equations in theplan been used only once?• Is each step of the mathematicslegitimate?Check each term for thecorrect units.• After all the substitution forunknowns, is the only unknownleft the target quantity?• Are the units the same on bothsides of the equation?Compute the value for thetarget variable andanswer original question.• Which values (numbers with units)from the physics descriptionshould be put into the equationfor the target quantity?• Do you need to convert units?Evaluate the SolutionFigure 5.2e. Flow Chart of Step 5 -- Evaluate the Solution

Chapter 5: Reinforcing Student Use of a Problem-solving Framework 67Execute the PlanEvaluate the SolutionCheck that answer isproperly stated.• Do the units make sense?• Do vector quantities have bothmagnitude and direction?• If someone else read just your answer,would they know what it meant?yesIs the answerreasonable?NoReview problem solution;make correctionsDetermine if the answeris complete.• Does the answer fit with your picture ofthe situation?• Is the answer the magnitude that youwould expect in this situation?• Do you have any knowledge of a similarsituation that you can compare with tosee if the answer is reasonable?• Can you change the situation (and thusyour equation for the target quantity) todescribe a simpler problem to whichyou know the answer?• Is your physics description complete?• Are the definitions of your physicsquantities unique?• Do the signs of your physics quantitiesagree with your coordinate system?• Can you justify all of the mathematicalsteps in your solution execution?• Did you use units in a consistentmanner in your execution?• Is there a calculation mistake in theexecution?• Have you answered the question fromthe Focus the Problem step?• Could someone else read and follow thesolution plan?• Are you sure you can justify eachmathematical step by referring to yourplan?A Good Solution

68 Part 1: Teaching Physics Through Problem SolvingAn example worksheet for theCompetent Problem-solvingFramework for an algebra-basedcourse is shown in Figure 5.3.We require our students to solveall individual and group testproblems and group practiceproblems on these sheets for thefirst 4 - 6 weeks of the course. 1Students can make copies ofthese sheets for solving theirhomework problems to helpthem practice correctly.As students become more comfortable with the problem-solving framework, theworksheets are dropped -- students write their solutions on blank paper. However,the steps are given on all test Information sheets, as illustrated in Chapter 3, page32.Instructor Problem SolutionsPosting your problem solutions to homework or test problems using the frameworkalso reinforces the student use. Figure 5.4 shows an example of a problem solutionfor a typical textbook homework problem. The problem is suitable for use at theend of the treatment of dynamics for students in an introductory algebra- orcalculus-based physics class. The details within a solution depend on the specificgoals of the physics class. In this case the solution was for an algebra-based class.Putting Together the Scaffolding for Students: AResource BookWe found it useful (and saves departmental funds) to put all the scaffolding forstudents into one resource booklet that students buy with their textbook or isavailable for download on the class web page. 1 This booklet includes the followingitems. An explanation of each step of the problem-solving framework. A one-page outline of the framework. Flow-charts for each step in the framework. A blank worksheet with cues from the framework (Figure 5.3). Students canphotocopy the worksheet for practice.

Chapter 5: Reinforcing Student Use of a Problem-solving Framework 69The remainder of the booklet is divided into three parts, one for kinematics, one fordynamics using Newton’s second law, and one for the conservation of energy. EachFigure 5.3. Answer Sheets for Competent Problem-solving FrameworkCalculus-based CourseFOCUS on the PROBLEMPicture and Given InformationQuestion(s)ApproachDESCRIBE the PHYSICSDiagram(s) and Define QuantitiesTarget Quantity(ies)

70 Part 1: Teaching Physics Through Problem SolvingQuantitative RelationshipsPLAN the SOLUTIONConstruct Specific EquationsEXECUTE the PLANFollow the PlanCheck UnitsCheck for SufficiencyCalculate Target Quantity(s)Outline the Math SolutionEVALUATE the ANSWERIs Answer Properly Stated?Is Answer Unreasonable?Is Answer Complete?

Chapter 5: Reinforcing Student Use of a Problem-solving Framework 71

72 Part 1: Teaching Physics Through Problem SolvingFigure 5.4. Example of a textbook problem solution on a worksheet.Problem: A 20-Kg block is pulled along a horizontal table by a light cord that extends horizontallyfrom the block over a pulley attached to the end of the table, and then down to a hanging 10-Kgblock. The coefficient of friction between the 20-Kg block and the table surface is 0.40. Determinethe speed of the blocks after moving 2 meters. They start at rest.FOCUS on the PROBLEMPicture and Given Information M = 20 kg voA = voB = 0am = 10 kg vfA = vfB = vV oB = 0Vd = 2 maA = aB = constantMBQuestion: Find speed of blocks after moving 2dm from rest.V oA = 0Approach: Assume a massless string and aa massless and frictionless pulley. Since theblocks move together, they always have thesame speed and magnitude of acceleration. Usem AV kinematics to relate final speeds toacceleration. Use Newton’s Laws to find theconstant acceleration.DESCRIBE the PHYSICSDiagram and Define VariablesMotion Diagram of Block Baav+xX oB = 0 X fB = 2 m = dt oB = 0 t f = t =?V oB = 0 V fB = ?Motion Diagram of Block Aa y oA = 0t o = 0V oA = 0Block AaT AW AT AW ABlock B aT BfN W B+yNf+xTW B B+yavy fA = 2 m = dt f = t = ?v = ?Constraints+y

Chapter 5: Reinforcing Student Use of a Problem-solving Framework 73Target Variable(s): v fA = v fB = VPLAN the SOLUTIONFind v: for Block Bv vvoBB 2 v B v 2Find v B : for Block Bv B x fB x oBt fB t oB v B d tFind t: for Block Ba B v fB v oBt fB t oBUnknownvv B a v taFind a: for Block AF A ma AW A T A ma mg T ma TFind T: for Block B (x direction) F xB Ma xB T f Ma fFind f f N NFind N: for Block B (y direction) F yB Ma yB NMg 0Check for Sufficiency:Yes -- 7 equations and 7 unknowns;solve problem working backwardsfrom to tQuantitative Relationships:a r v fr v ort f t ov r r f r ot f t ov r v fr v or2 F r ma rf NW mgEXECUTE the PLANSolve for N and put into ;NMg 0N Mgf MgPut into and solve for T:T Mg MaT Ma MgPut into and solve for a:mg (Ma Mg) mag(m M) a(m M)g m Mm M aPut into and solve for t: Put into :a v tt v av B d tv(m M)g(mM)dg(m M)v(m M)Put into and solve for v:v B v dg(m M)2 v(m M)v 2 2dg(m M)m Ma A a B av fA v fB vT A T B T 2 2m9.8m /s2 (10kg - 0.40kg)10kg+ 20kg 2.6 m 2 / s 2v = 1.6 m/s

74 Part 1: Teaching Physics Through Problem SolvingEVALUATE the ANSWERDoes the mathematical result answer the question asked? YES, the speed of the blocksafter moving 2 meters from rest is 1.6 m/s.Is the result properly stated with appropriate units? Yes, the units of v are m/s.Is the result unreasonable? NO. v should be less than for free fall (6.3 m/s), and it is.section starts with an explanation of how to draw the appropriate physics diagrams:motion diagrams, force diagrams, and energy tables. Each section also includes thefollowing. Several textbook like problems with solutions on the worksheets. Several context-rich problems for students to practice solving using theproblem-solving framework.Endnotes1 For example, see Heller, K. & Heller, P. (2000), The competent problem solver forintroductory physics: Algebra, McGraw-Hill Higher Education; and Heller, K. & Heller,P. (2000), The competent problem solver for introductory physics: Calculus, McGraw-HillHigher Education

75Chapter 6Why Cooperative Problem Solving:Best PracticesIn this chapter: Why use cooperative problem solving – the solution to a dilemma. A theoretical framework,. Establishing an environment of expert practice The relationship between course structures and cognitive apprenticeshipChapter 2 addressed why many students in introductory courses fail tolearn physics through problem solving. In Chapters 3 and 4 we claimedthat students need to solve complex questions, such as context-richproblems, using a problem-solving framework that emphasizes making decisionsbased on physics concepts. Unfortunately, introducing suitable problemstogether with an expert-like problem solving framework, as outlined in Chapter5, is not enough for students to be successful in learning physics throughproblem solving. In this chapter we describe the dilemma that led us to adoptCooperative Problem Solving (CPS).For many of us it is useful to have a theoretical framework to guide our actions,whether in science or in teaching. The final section of this chapter describes thetheoretical framework we found useful implementing a coherent instructionalstrategy for introductory courses that leads to the improvement of students’conceptual understanding, problem solving performance, and more expert-likeattitudes towards physics and learning physics. To be fair, we developed muchof this pedagogy empirically before this theory was available based on goodteaching practice and observation of student behavior. Nevertheless, once wewere aware of the theory, the role played by the features of that pedagogy andwhat additional features should be added.The Dilemma and a SolutionTo learn physics, students need to examine their personal physics ideas and howthey apply them in different situations. Problem solving, requires students to dojust that if the questions they are addressing are complex, sophisticated andstraightforward enough enough, such as context-rich problems, that requiresusing a framework that emphasizes making decisions using physics concepts.This problem-solving framework is initially unfamiliar, complex, and unnatural

76 Part 1:Teaching Physics Through Problem Solvingfor beginning students. Even after clear and careful instruction, most studentswill fail in their initial attempts to use it. As a result they become frustrated, andrevert to their familiar novice strategies (see Chapter 2 for novice strategies)which while they may not succeed are at least familiar.It is reasonable to think that it might begin having them practice this problemsolving framework with simpler, more straightforward problems such as thosefound in many textbooks. We did. We thought students could practice theproblem-solving framework on those problems until the framework becomemore familiar, then they could successfully go on to context-rich problems.Unfortunately those straightforward problems yield, or seem to yield, to thestudents’ novice strategies. Students see no reason to apply an unfamiliarproblem-solving framework. They did not get the practice. That gave adilemma:‣ If students are given context-rich problems to solve usinga competent problem-solving framework, then they areinitially unsuccessful. There is no reason to change fromtheir novice strategy that is also unsuccessful, but at leastfamiliar.‣ If students are given “easy” problems to solve using acompetent problem-solving framework, they are initiallysuccessful using a novice strategy. Again, there is nomotivation to change.A solution to this dilemma is to use cooperative groupsto provide support, so that students can be initiallysuccessful solving context-rich problems using an expertlikeframework that has been explicitly demonstrated inclass. The practice with cooperative groups provides thecoaching that students need to move toward to an expertlikeframework when solving problems individually. Howthis is done is discussed in Parts 2 and 3.A Framework for Teaching and Learning IntroductoryPhysics: Cognitive ApprenticeshipA useful instructional theory should help make connections between as manydifferent aspects of teaching and learning as possible. Of course, it should agreewith the currently available data and have some predictive power. As in science,it is not necessary that a theory be true, in some absolute sense, for it to beuseful. The theory that we find useful in teaching introductory physics is calledcognitive apprenticeship (or situated learning). 1 It supports out 1st Law ofInstruction: Doing something once is not enough.In a very real sense, the cognitive apprenticeship theory provides a description

Chapter 6: Why Cooperative Problem Solving 77and justification of the very familiar educational scheme that is the basis for ourgraduate education. The theory also provides very practical guidance forteaching classes.Cognitive Apprentice is based on two observations.‣ Learning something like physics is a complex process that depends onstudents’ existing knowledge and how they use that knowledge. Learningdepends on the unique background of each learner.‣ Apprenticeship is the most effective type of instruction that humans havedevised for complex learning.The goal of the inventers of cognitive apprenticeship was to identify theessential features that make apprenticeship so powerful and apply them to theclassroom situation. One key feature of apprenticeship is that learning isdirectly connected to a situation that is meaningful to the student. Anotheressential feature is that the student must observe the action of experts, and theresults of that action, from beginning to end. The beginning of an action musthave a motivation meaningful to students (i.e., who cares?). The end of anaction is a conclusion that students perceive as useful (i.e., what good is it?).Within this framework, students must immediately engagein tasks that, while at a beginning level, they perceive asfitting into a more complex expert-like task. As studentsget such experience in a variety of contexts they buildneural pathways that connect the new knowledge to theirexisting experiences. This integration makes newknowledge accessible to students, and is more useful thaninformation learned in isolation. A rich context forcesstudents to examine and perhaps modify their existingknowledge,, expressed as neural connections, whenachieving a desired result.Students begin with a knowledge structure that is a complex and unknownnetwork of neural connections (see Chapter 2, page //). They can only explorethese networks while in the act of using (activating) them and determining ifthey have achieved the desired results. Successful learning requires theestablishment of new links in and between existing networks, establishing newnetworks, and eliminating or weakening old links in or between existingnetworks. This learning can be facilitated by instruction that is rich enough incontext so that it is individualized, even in a large class. For humans, thisconstruction of individual knowledge is fostered in large part, by interactionswith other people. 2In an apprenticeship system, teaching has four essential functions: modeling,scaffolding, coaching, and fading. That is, first show students in detail what youwant them to do (model or demonstrate). Provide structure or support for

78 Part 1:Teaching Physics Through Problem Solvingstudents to practice using what you want them to learn (scaffolding). Havestudents practice in their own way what you have demonstrated with correctivefeedback as they are doing it (coach). Have them practice performing the taskthemselves while withdrawing some of the scaffolding (fade). This is not asequential process in which each step is performed only once. Effective learningrequires looping through these functions as needed by the individual student.Each of these functions is described below.ModelingLearning to accomplish a complex task is most efficient if the entire task is firstshown to the students. Organizing the information from that demonstration ofthe process is aided if the learner knows the motivation for the task and majorsubtasks that comprise that task. For example, the first step in learning how toplay baseball is to see a baseball game and to have the motivation for the majoractions, such as hitting or running the bases, explained. Think how difficult itwould be for people to learn how to play a game of baseball if they were firsttaught the vocabulary (e.g., base, ball, home run, double), then coached inspecific skills (e.g., batting, pitching, sliding, bunting), all before they saw agame of baseball or were given the opportunity to play the game. Of course,having seen one or even many baseball games does not mean that you can playbaseball.For a student to learn an intellectual process,such as using the fundamental concepts ofphysics to solve problems, they must first seehow the game is played. Someone must showthem problems solved from their beginning totheir end while making sure that all of theinternal intellectual subtasks are explicitlyshown to the student. Because people’sobservations are processed through their existing knowledge structure, suchdemonstrations of the same process must be repeated as the student gainsknowledge. An experienced baseball player’s observations of a baseball gameare quite different from those who have never seen the game before, but bothlearn from the experience.Even after a task is demonstrated in a clear and detailed manner, most peoplecannot accomplish that task on their own. They need to practice. However,practice is only beneficial if the correct procedures are practiced. A novice willusually practice an idiosyncratic blend of the new thing they were expected tolearn and their existing knowledge base. This type of practice can actually bedamaging to their progress. As a famous and successful football coachremarked, “Practice does not make perfect, only perfect practice makes perfect.”[reference Vince Lombardy] Every learner needs scaffolding and an opportunityto practice while getting immediate feedback. A coach provides this feedback.Modeling is a necessary but not sufficient condition for learning -- scaffolding

Chapter 6: Why Cooperative Problem Solving 79and coaching are essential. 3ScaffoldingIn architecture, scaffolding is a temporary structureworkers use during the construction of a building.Scaffolding lets workers reach places that wouldotherwise be inaccessible. In education, scaffolding isstructure that supports learning. Scaffolding caninclude helpful instructor comments, a compelling taskor problem, templates such as worksheets for problemsolving, practice problems, grading rubrics, andcollections of useful resources. 4 An example ofscaffolding is training wheels for learning to ride abicycle, that is taken away over time to assure that thelearner is able to perform the skill on their own.We have provided many examples of scaffolding in this book. Context rich problems (Chapters 3, 8, 15, and Appendices B and C) Verbal description of a problem-solving framework (Chapters 4 and 5) Problem-solving flow charts (Chapter 5) Information sheets for tests (Chapters 3 and 11) Worksheets with problem-solving cues (Chapters 5) Instructor solutions (Chapters 5 and 11) Description of roles for cooperative problem solving (Chapter 8) Forms for groups to evaluate how their functioning while solving agroup problem (Chapters 8 and 11)In our introductory physics courses, the first six examples of scaffolding are alsoprovided in a resource book for students.Scaffolding and theThird Law of Education.You may have noticed thatscaffolding is related to ourThird Law of Education: Make iteasier for students to do what you wantthem to do (ladders), and moredifficult to do what you don’t wantthem to do (fences). Scaffoldingcomprises the ladders up thelearning mountain.

80 Part 1:Teaching Physics Through Problem SolvingCoachingPeople learn by doing, but they require guidance to correct faults when theyattempt to become skillful at a task that has been demonstrated. Without thisguidance, practice can reinforce bad habits. “Coaching” consists of helpfulinstructor clues, reminders, or any type of constructive feedback given tostudents while they are practicing a complex task such as problem solving in their own way.Coaching gives students immediate feedback when they are most aware of theinterconnections of their problem-solving actions to their knowledge base.Coaching begins from an individual student’s approach. It is not a repeatedillustration of the instructors “correct way.”. Coaching can only happenindividually or in a small group because each person has a different backgroundwith different experiences, learning styles and talents (see Chapter 13 for how tocoach in CPS). An experienced coach can diagnose a student’s difficulties with atask, probe the student’s knowledge network as applied to accomplishing thattask, decide on a treatment for those difficulties, and guide the student inresolving the difficulties. One important part of coaching is to guide students’actions by structuring the situation in which they practice.In higher education, three different kinds of coaching can usually be provided tostudents: coaching by an expert (instructor), coaching by a person moreexperienced than the student (graduate or undergraduate teaching assistants),and peer coaching by fellow students at the same level.Expert Coaching. An experienced coach canobserve a student’s actions and evaluate what thatstudent is attempting to do, why they are probablyhaving difficulty, and how the student might makeprogress based on expert procedures and thecurrent ability of the student. This coach canselect from among several techniques guide thestudent to one that is appropriate. This coach canalso construct artificial situations to point outinconsistencies in the student’s thinking or constrain the student to not practicea subtask incorrectly.Experienced Coaching. A person with more experience than the student inaccomplishing that task can diagnose a student’s ineffective or incorrectprocedures. This coach can point out inconsistencies in the student’s thinkingand demonstrate a correct procedure usually based on personal experience.Peer Coaching. Peers can question whetheran action is being done correctly or ineffectivelyand show how they do the action. In addition, topointing out inconsistancies in the knowledgestructure of others, the act of explaining is itselfone of the most effective methods for learning.

Chapter 6: Why Cooperative Problem Solving 81Each type of coach has advantages in an instructional situation. Clearly, thepeer coach is more closely attuned to the vocabulary, meaningful examples, andways of thinking of the other students and is less intimidating. Students aremore likely to try out their own ideas with a peer than with a professor, or evenwith a more advanced student. Peer coaching is very efficient because everyoneis in the process is learning. Every instructor knows that the best way to learnsomething is to teach it. However, peer coaches often do not have sufficientknowledge to recognize an error, or the experience to suggest how to make anidea plausible to others.On the other hand, the expert coach can more easily recognize students' errorsand suggest lines of thought that may be more aligned with those of thestudent.. Because the expert is usually also an authority figure, this coach cansuppress a student’s necessary exploration of their own ideas. With an expertcoach, the student almost never gets the learning advantage of being the teacher.Experienced coaches are a compromise between expert and peer coaches. Theybring more expertise about the subject matter and different pathways to asolution than a peer coach. They can also be less of an authority figure than theexpert coach. As a practical matter, there are usually more peer coachesavailable than either expert or experienced coaches. For this reason alone, peercoaches are useful to give the instant feedback so necessary in the coachingprocess.A multi-level coaching system seems to work best in situations with a widevariety of students. The expert coach sets the practice task and designs thelearning situation under which the task is carried out. The peer coaches exploretheir own, and each other's, way of accomplishing the practice task guided bytheir collective impressions of the correct outcome from the previousdemonstrations of the process. The experienced coach corrects mistakes notcaught in the peer coaching process and guides the students to be more effectivepeer coaches. In a traditional apprenticeship the master, the journeyman, andthe other apprentices accomplish these three levels of coaching. In a researchgroup the professor, the post doc and advanced graduate students, and the othergraduate students at the same level accomplish the three levels of coaching.FadingThe goal of instruction is for students to be able toaccomplish a task on their own and in situationsdifferent from that of the instruction. Fading is theact of slowly removing as much of the instructionalguidance as possible. Not only must the teacher setup the instructional framework to facilitate thecoaching function, but must also remove much ofthis framework before the end of the course. Forexample, if worksheets are introduced to guide students through a problemsolvingframework, they need to be replaced by plain paper before the student

82 Part 1:Teaching Physics Through Problem Solvingleaves the course. Another example is that students need to solve problems ontheir own in addition to solving them in a group.ConsistencyResearch indicates that instruction that incorporates one or more commonfeatures contributes to better problem-solving performance in students. 5 Thesefeatures are: (1) explicit teaching of problem-solving heuristics (similar to theCompetent Problem Solving Framework); (2) modeling (demonstrating) the useof the heuristics by the instructor, and (3) requiring students to use the heuristicsexplicitly when solving problems. These are three features of cognitiveapprenticeship. [what about coaching???]Establishing an Environment of Expert PracticeIn the learning model called cognitive apprenticeship the actions describedabove must be carried out in what is called an environment of expert practice.An environment of expert practice exists when each student knows what theinstruction is trying to accomplish and why it is important to the student. Inother words, every student should be able to answer the following threequestions at any time in the course:1. Why is what we are learning now important to me?2. How is it related to what I already know?3. How might I use this knowledge?Constructing the environment of expert practice in a classroom requires that theinstructor begin a lesson with a motivation within a context that is perceived asmeaningful and useful to the student. It must continually and explicitly pointout how the parts of the lesson link to students’ previous knowledge, includingtheir experiences. Because students do not have the same motivations orexperiences, establishing an environment of expert practice means using avariety of contexts, examples, and motivations throughout the lesson. A lessonshould not be structured as a mystery story with a surprising or interesting revealat the end.

Chapter 6: Why Cooperative Problem Solving 83Course Structures and Cognitive ApprenticeshipWithin a traditional course structure, modeling can be done in the lecture part ofthe class and coaching in the discussion and/or laboratory sections with the helpof cooperative groups. In a studio or laboratory based course, modeling andcoaching of problem solving can be interweaved as necessary in a very effectivemanner. Fading occurs in all venues, but is particularly apparent in individualassignments such as laboratory reports and on tests.Lectures and Demonstrating the Problem-solvingProcessThe lecture is an effective method for demonstratingproblem solving by showing, explaining, and motivatingthe details of each step of the solution process. Onecan introduce new physics topics by attempting to solvethat needs the development of a new concept for itssuccessful resolution. Demonstrating the problemsolvingprocess during lectures can actively engagestudents. It is always important to allow the studentsseveral minutes to read the problem and begin their ownsolution before the demonstration of the processbegins. This process serves as an advanced organizer sothat students begin to access that part of their knowledge network that isrelevant. To incorporate some peer coaching, students can be encouraged tocompare their start of a solution with those of their neighbors and try to resolveany differences.Good educational practice has always suggested pauses in a lecture to askstudents a simple question and allow them several minutes to answer in writing,or now electronically. That question may be an elaboration of a step in theproblem-solving process, a simple application of a point just demonstrated, orthe logical next step in the solution process. Again peer coaching can beintroduced by have students compare their results with their neighbors even in alarge lecture class. To get good student participation, remember the Zeroth Lawof Instruction (If you don't grade it, students don't do it) and collect at least someanswers for grading. Electronic techniques using “clickers” makes this easy todo.Demonstrating every decision in a problem-solving framework takes time. Youwill not be able to go through many problems in a class period. However, whenyou demonstrate problem solving that begins with the fundamental principles(e.g. Newton’s second law, conservation of energy), every problem solved toillustrate one concept also reinforces problem solving for other concepts.Remember that the purpose of demonstrating is to illustrate the problem-solvingprocess. Students will still need coaching to actually be able to do it.

84 Part 1:Teaching Physics Through Problem SolvingLectures and CoachingIt is possible to have some peer coaching interweaved into lectures. Whenpausing a lecture to ask questions, and after each student has written anindividual answer, ask them to compare their answers with their neighbors. Thistime honored technique of peer coaching is especially effective just before ananswer is to be submitted for grading. 6 The few minutes necessary for thistechnique is remarkably effective in keeping students involved in the processwhen the lecturer demonstrates the construction of a problem solution. SeeChapter 9 page 102 for an outline of steps for demonstrating and coaching theuse of a problem-solving framework for solving problems.Sections and CoachingThe most effective coaching occurs in a small classroom situation that isphysically configured to facilitate students working together with both peer andexperienced coaching (see Chapter 9, pages 103 - 106). Cooperative grouping 7 isa very successful technique to structure thecoaching process that has been used from elementary schools to businesssettings. It has been applied in many subject areas in the university. Describingcooperative grouping and its advantages for coaching problem solving inintroductory physics is discussed in Part 3 (Chapters 11, 12, and 13).Endnotes1See, for example: Collins, A., Brown, J.S. & Newman, S.E. (1989), Cognitiveapprenticeship: Teaching the crafts of reading, writing and mathematics, inKnowing, learning, and instruction: Essays in honor of Robert Glaser edited by L.B.Resnick, Hillsdale NJ, Lawrence Erlbaum, pp. 453–494; and Brown, J. S.,Collins, A., & Duguid, P. (1989), Situated cognition and the culture oflearning, Educational Researcher, 18(1), 32-42.2 See, for example, the research summary in Bransford, J., Brown, A., & Cocking, R.(Eds), (2000), How people learn: Brain, mind, experience, and school, Washington, DC:National Academy Press. Cognitive Apprenticeship is often associated with thesocial learning theory of Lev Vygotsy. See, for example, Vygotsky, L. S. (1978).Mind in society: The development of higher psychological processes. Cambridge, MA: HarvardUniversity Press.3 Many researchers refer to two types of scaffolding, soft or contingent scaffolding(such as cooperative grouping with feedback) and hard scaffolding (e.g., problemsolving flowcharts. See, Saye, J.W. and Brush, T. (2002), Scaffolding criticalreasoning about history and social issues in multimedia-supported learningenvironments, Educational Technology Research and Development, 50(3), 77-96

Chapter 6: Why Cooperative Problem Solving 854567Scaffolding has been defined and discussed theoretically in a variety of differentways. See, for example, Hartman, H. (2002). Scaffolding and cooperativelearning, Human Learning and Instruction. New York: City College of CityUniversity of New York; Zydner, J. M. (2008). Cognitive tools for scaffoldingstudents defining an ill-structured problem, Journal of Educational ComputingResearch, 38(4), 353-385; Saye, J.W. and Brush, T. (2002), Scaffolding criticalreasoning about history and social issues in multimedia-supported learningenvironments, Educational Technology Research and Development, 50(3), 77-96;Jonassen, D. (1997), Instructional design models for well-structured and illstructuredproblem-solving learning outcomes. Educational Technology Research andDevelopment, 45(1), 65-94; Holton, D. and Clark, D. (2006), Scaffolding andmetacognition, International Journal of Mathematical Education in Science and Technology,37, 127-143.Taconis, R., Ferguson-Hessler, M.G.M., &. Broekkamp, H. (2001), Teachingscience problem-solving, Journal of Research in Science Teaching, 38, 442–468. Theauthor did a meta-analysis of 22 previously published articles on teachingproblem solving in science classes. They also found that having students work ingroups did not improve problem solving unless the group work was combinedwith the teaching of problem-solving heuristics, modeling the use of the heuristicsby the instructor, and/or requiring students to use the heuristics explicitly whensolving problems.McKeachie, J.W. (1996), Teaching tips: strategies, research, and theory for college anduniversity teachers, Lexington, MA, D.C. Heath. For an application in physics seeMazur, E. (1996), Peer Instruction: A User’s Manual, Prentice-Hall.Johnson, D.W, Johnson, R.T. & Smith, K.A. (2006), Active learning: Cooperation inthe college classroom, 3 rd Ed. Edina MN, Interaction Book Company.

86 Part 1:Teaching Physics Through Problem Solving

Part87

88In this part . . .There is an old cliché that fits here, “If it ain’t broke, don’t fix it!” We areassuming that while your introductory physics course may not be completelybroken, you are dissatisfied with your students’ problem-solving performanceand are looking for a way to “fix it.” There are many physics educationreforms from which to choose. In the chapters of Part 2 we provideinformation to help you decide whether you want to adopt CooperativeProblem Solving.This part of the book attempts to answer two questions to help youdetermine if cooperative problem solving might be useful in your course.The first question is: What is cooperative problem-solving (CPS)? Inanswering this question, Chapter 7 describes the differences betweenstudents doing group work and students working in a cooperative group.Chapters 8 and 9 give practical information about the second question: Whatcourse changes are needed for optimal implementation of CPS? Chapter 8discusses appropriate problems and how to structure and managecooperative groups for optimal implementation of CPS. Chapter 9 discusseshow to structure a course for CPS, including scheduling and other resources(personnel and space), as well as appropriate grading practices for thecourse and for grading students’ problem solutions. Both chapters convey“what to shoot for,” and not where you can make a reasonable beginning.The last section Chapter 9 outlines a way to get started in lecture withinformal groups.Chapter 10 describes the research results for improvement in problemsolving skills and conceptual understanding of physics with CooperativeProblem Solving (CPS), both for partial and full implementation.

89Chapter 7Cooperative Problem Solving:Not Just Working in GroupsIn this chapter: The differences between traditional and cooperative groups. The five elements of cooperative problem solving. The differences in achievement in traditional and cooperative settings.Just what is cooperative problem solving? The answer to this question is nothaving students spend some time each week solving problems in a group. Wehave often met instructors who have had students solve problems in groups,with no improvement in individual problem-solving performance. In fact theresearch to date indicates that group work, by itself, does not raise theachievement of students. As is often the case, to understand what cooperativegroupproblem solving is, it is helpful to know what it isn’t. In this chapter wedescribe some of the differences between traditional-group and cooperativegroupproblem solving.Two ExamplesImagine that you observed two small classes in which students work in groupsto solve a problem. Each class consists of 15 – 18 students, and they are solvinga kinematics problem with two-dimensional motion. The first class is what wewill call “traditional-group problem solving.” In the second class, students areengaged in cooperative-group problem solving. Below is a description of whatyou might observe in each class.Example of Traditional-group Problem SolvingBefore class begins, the students are sitting quietly inrows facing the instructor. The instructor begins theclass by talking for about 15 minutes, reviewing theprojectile motion equations, then tells the students toget into groups and solve Problem #5 at the end ofChapter 2 in their textbook. The instructor sits at adesk in front of class.The students get into groups, usually with their neighbors. The class is veryquiet, and most students are working independently. They flip through Chapter 2 for awhile, then settle on a page what seems to have a relevant example solution to asimilar problem. Each student starts writing a solution. Occasionally they talk

90 Part 2: Using Cooperative Problem Solvingwith each other, asking questions about what page they are on or what equationto use.After about 20 minutes, some students turn towards eachother and start comparing their solutions. They start bycomparing their numerical answers. In the groups wheremembers got the same answer, the conversation usuallyturns social. In the groups with different individualanswers, students compare the equations they used andthe numerical answers for different intermediatequantities. In some groups, a student will recognize oragree that they did something wrong, and decide tochange their solution. In other groups, however, students do not come to anagreement about how to solve the problem.The instructor goes from group to group, answering the individual questions ofstudents. Shortly before the end of class, the instructor collects the students'individual solutions and shows them on the wall board how to solve theproblem.Example of Cooperative Problem SolvingJust before class students are clustered in groups of three or four, facing eachother and talking. The instructor begins class by talking about 5 minutes --reminding students that they have just started two-dimensional motion, that theproblem they will solve today was designed to help them understand therelationship between one-dimensional and two-dimensional motion, and theywill have 35 minutes to solve the problem. The instructor gives one person ineach group (the recorder) and answer sheet, and provides all students access tothe problem and all the fundamental equations they have studied in class to thispoint in time (either projected or on individual sheets).The class is quiet for a few minutes as they read theproblem, then there is a buzz of talking for the next 30minutes. No textbooks or notes are open. Groupmembers are talking and listening to each other, andonly one member of each group is writing a problemsolution. They mostly talk about how the objects aremoving, what they know and don’t know, what theyneed to assume, the meaning and application of theequations that they want to use, and the next steps theyshould take in their solution.The instructor circulates slowly around the room, observing andlistening, occasionally interacting with the groups that theinstructor judges need the most help. This pattern of circulatingand intervention continues for about 30 minutes. At the end of

Chapter 7: Cooperative Problem Solving: Not Just Working In Groups 91that time, the instructor assigns one member from each group (usually not themember who recorded the solution) to draw a motion diagram and write theequations they used to solve the problem on a wall board. That member can askfor help from the remaining group members as necessary. The instructor thentells the class to examine the board for a few minutes to determine thesimilarities and differences of the group solutions. Based on what is on the wallboard, the instructor then leads a class discussion focused on the confusionbetween the perpendicular components of two-dimensional motion.These examples illustrate several differences between traditional group problemsolving and cooperative problem solving. These differences are summarized inFigure 7.1. Cooperation is not having students sit side-by-side and talk witheach other as they do their individual assignments. Cooperation is not assigninga problem (or lab report) to a group of students where one student does all thework and others put their names on it. Cooperation is not having students solveproblems individually with the instruction that the ones who finish first are tohelp the slower students. Cooperative Problem Solving is much more thanbeing physically near other students, discussing the problem with other students,helping other students solve the problem, or sharing procedures, although eachof these is important.Elements of Cooperative Problem SolvingJohnson, Johnson, and Smith (2006) 1 characterize cooperative learning throughfive basic elements. These elements are described briefly below.Positive Interdependence exists when eachstudent in a group believes they are linked such thatthey cannot succeed unless everyone contributes.In other words, they believe that they “sink or swimtogether.”In a problem-solving session, positiveinterdependence is structured by group members:(1) agreeing on the answer and solution strategies(goal interdependence); and (2) fulfilling assignedrole responsibilities (role interdependence). Theinstructor must assess the group problem solutions and give the results back tothe groups. Occasionally the result of the assessment is a grade, with eachgroup member getting the same grade (reward interdependence). See Chapters 8and 9 for more details about role assignment and grading.

92 Part 2: Using Cooperative Problem SolvingFigure 7.1. Differences between traditional group and cooperative group problem solvingTraditional GroupsStudents are assigned to worktogether, and accept that they have todo so (or at worst, have no interest indoing so).The group problems require very littlejoint work -- they can be solved aseasily by an individual as by a group.Cooperative GroupsStudents are assigned to work togetherand recognize that it is useful to do so.The group problems require a level ofphysics knowledge and decisionmakingthat make them difficult foreven the best students' to solveindividually.Best Case: Students believe that theywill be evaluated and rewarded asindividuals, not as group members. Sothey interact primarily to compare theirprocedures and solutions. There is nomotivation to teach or learn from eachother.Worst Case: Students believe they arecompeting for the best grades. Sowhen they interact, they view eachother as rivals who must be defeated or"dummies" whose questions steal timefrom their own learning.Students believe that their successdepends on the efforts of all groupmembers. They interact by coconstructinga single problem solution.They help each other by clarifying,explaining, and justifying ideas andprocedures as they solve the problem.When groups are not functioning well,individuals either disengage or blametheir other group members for notbeing prepared. Students who are notdoing as well as they would like blamethe other group members for “draggingthem down.”Students hold each other accountablefor doing high quality work while theysolve the problem together. Allmembers take responsibility forproviding leadership and resolvingconflicts.After the review, the instructor doesnothing except answer factualquestions (or at worst, spends all thetime helping a few students get theright answer).The instructor constantly monitorsgroups, coaching groups that needhelp with specific points of physics andgroups that are not functioning well.This is followed by a whole-classdiscussion.

Chapter 7: Cooperative Problem Solving: Not Just Working In Groups 93Face-to-Face Promotive Interaction exists when eachstudent orally gives a provisional approach to a task,discusses with other members of the group the natureof the physics concepts and problem-solving techniquesbeing used, teaches their knowledge to the other groupmembers, and discusses the connections betweenpresent and past learning. This face-to-face interactionis called “promotive” because students promote (i.e., help, assist, encourage, andsupport) each other’s efforts to learn.Individual Accountability/Personal Responsibilityrequires the instructor to ensure that the performance of eachindividual student is assessed and the results given back to theindividual as well as the group. Each student needs to knowthat they are ultimately responsible for their own learning. Noone can “hitch-hike” on the work of others. Common waysof structuring individual accountability include givingindividual exams, choosing a group spokesman at randomwhen the instructor interacts with a single group and when thegroup interacts with the class (see Chapter 12, page //).Collaborative Skills are necessary for groupfunctioning. Students must have and use the neededleadership, decision-making, trust-building,communication, and conflict-management skills.Explicit attention must be paid to the development ofthese skills, especially with students who have neverworked cooperatively in learning situations. Acommon way to reinforce collaborative skills isthrough role assignment (see Chapter 8, pages //-//) and coaching in theseskills when intervening with dysfunctional groups (see Chapter 13).Group Processing requires each group to discussamong its members how effectively they workedtogether and maintained an effective workingrelationship. For example, at the end of their workingperiod the groups might discuss their functioning byanswering two questions:1. What is something each member did that washelpful for the group?2. What is something each member could do to make the group more effectivenext time?Such processing is a quick method of diffusing resentments, improvingcollaborative skills, ensuring timely feedback, and reminding students thateveryone must contribute for the group to be efficient (see Chapter 8, pages //to //).

94 Part 2: Using Cooperative Problem SolvingFigure 7.2. A representation of the performance level of traditional and cooperative groups:a summary of research across different age groups and subject areas.The next five chapters explain in detail how these elements of cooperativelearning can be implemented in Cooperative Problem Solving (CPS).Achievement for Traditional Groups and Cooperative Groups:SummaryThe major difference between traditional-group and cooperative-group problemsolving is how students go about solving the problem. In traditional-groupproblem solving, students tend to solve the problem individually, typically withthe help of a textbook or class notes, then compare their answers in groups. Incooperative group problem solving, students are actively engaged in the coconstructionof a problem solution, based on what they know at the time.There are several learning advantages to the collaborative co-construction of aproblem solution. 2 During this process, students actively examine their ownconceptual and procedural knowledge they request explanations andjustifications from each other. This constant process of explanation andjustification helps clarify each member’s thinking about the physics concepts tobe used, and how these concepts should be applied to the particular problem. 3Moreover, each member practices and observes the thinking strategies of others.This introduces differences that they can incorporate into their problemsolutions.Nearly 600 experiments and over 100 correlation studies have been conducedduring the last 100 years comparing the effectiveness of cooperative,competitive, and individual learning environments. Cooperation amongstudents typically results in higher achievement and greater productivity for

Chapter 7: Cooperative Problem Solving: Not Just Working In Groups 95different age groups, in different subject areas, and in different settings. 4 Asummary of the studies conducted at the higher education level may be found inJohnson, Johnson, and Smith (2006).1From this extensive research, the difference in achievement of traditional-groupand cooperative-group problem solving is summarized in Figure 7.2. At theworst (when students compete for grades), group performance is less than thepotential of the individual members. The average class performance is typicallylower than in classes with no group work. At the best (when students areevaluated against an absolute standard), the group performance is better thanthat of some of the members but the more hard-working and conscientiousmembers would perform better if they worked alone. The average classperformance on individual tests is about the same or slightly higher than inclasses with no group work. With CPS, the group is more that the sum of itsparts, and all students (even the best students) perform better on both groupand individual problems than if there were no group component to theinstruction. The average class performance is higher than in classes with nogroup work. 5Endnotes1Johnson, D.W, Johnson, R.T. & Smith, K.A. (2006). Active learning: Cooperation inthe college classroom, 3 rd Ed. Edina MN, Interaction Book Company.2See, for example: Schoenfeld, A.H. (1983), Beyond the purely cognitive: Beliefsystems, social cognitions, and meta-cognitions as driving forces in intellectualperformance," Cognitive Science, 8, 173-190; Brown, A. L. & Palincsar, A. S.(1989), Guided, cooperative learning and individual knowledge acquisition, inKnowing, learning, and instruction edited by L. B. Resnick, Hillsdale NJ, LawrenceErlbaum, pp. 393-451; and Collins, A., Brown, J. S. & Newman, S. E. (1989),Cognitive apprenticeship: Teaching the crafts of reading, writing andmathematics," in Knowing, learning, and instruction: Essays in honor of Robert Glaser,edited by L. B. Resnick, Hillsdale NJ, Lawrence Erlbaum, pp. 453 – 494.3Hollabaugh, M. (1995), Physics problem-solving in cooperative learning groups, Ph.D.Thesis, University of Minnesota4See Brown, A. L. & Palincsar, A. S. (1989), Guided, cooperative learning andindividual knowledge acquisition, in Knowing, learning, and instruction edited by L.B. Resnick, Hillsdale NJ, Lawrence Erlbaum, pp. 393-451; Johnson, D.W.,Johnson, R.T., & Smith, K.A. (1991), Cooperative learning: Increasing college facultyinstructional productivity. ASHE-ERIC Higher Education Report 1991-4,Washington, DC: The George Washington University, School of Education andHuman Development; and Johnson, D.W., Johnson, R.T., and Smith, K.A.(2007), The state of cooperative learning in postsecondary and professionalsettings. Educational Psychology Review, 19 (1), 15-29.

96 Part 2: Using Cooperative Problem Solving5For data supporting these two statements, see Heller, P., Keith, R., & Anderson,S. (1992), Teaching problem solving through cooperative grouping. Part 1:Groups versus individual problem solving, American Journal of Physics 60(7), 627-636.

97Chapter 8Managing Groups andAppropriate Group ProblemsIn this chapter: Managing groups: group size, assignment, changing groups, role assignment and rotation, andgroup processing. Why textbook problems are ineffective group problems. What are the criteria for effective problems?Like a table, CooperativeProblem Soving (CPS) issupported by four legs: Managing groups for optimallearning. Appropriate group problems Appropriate grading Appropriate course structure forcooperative problem solvingIf one or more of the legs is short,then the table is unbalanced. Thelearning gains will be less than withoptimal implementation of CPS. How close to the optimal you can get depends onyour constraints and experience (see Chapters 9 and 11).This chapter contains suggestions and recommendations for and structuring andmanaging groups and appropriate group problems. The following chapter providesrecommendations for structuring the introductory physics course and appropriategrading.Group Structure and ManagementThere are several aspects of group structuring that affect learning, such as groupsize, group composition, how long groups stay together, and the roles of individualstudents in the groups. Our recommended structures and their rationale aredescribed in this section. The figures contain a brief description of our research thatsupports each structure. 1

98 Part 2: Using Cooperative Problem SolvingFigure 8.1. Why three-member groups are better than pairs or four-member groups?For the co-construction of a physics problem solution by students in introductory courses, wefound the "optimal" group size to be three members. A three-member group is large enoughfor the generation of diverse ideas and approaches, but small enough to be manageable sothat all students can contribute to the problem solution.An examination of written group problem solutions indicated that three- and four-membergroups generate a more logical and organized solution with fewer conceptual mistakes thanpairs. About 60 - 80% of pairs make conceptual errors in their solution (e.g., an incorrectforce or energy), whereas only about 10 - 30% of three-or four member groups make thesesame errors. Observations of group interactions suggested several possible causes for thelower performance of pairs. Groups of two did not seem to have the "critical mass" ofconceptual and procedural knowledge for successful completion of context-rich problems.They tended to go off track or get stuck with a single approach to a problem, which was oftenincorrect.With larger groups, the contributions of the additional student(s) allowed the group to jumpto another track when it seemed to be following an unfruitful path. In some groups of two,one student often dominated the problem solving process, so the pair did not function as acooperative group. A pair usually had no mechanism for deciding between two strongly heldviewpoints except the constant domination of one member, who was not always the mostknowledgeable student. This behavior was especially prevalent in male-female pairs. Inlarger groups, one student often functioned as a mediator between students with opposingviewpoints. The issue was resolved based on physics rather than the personality trait of aparticular student.In groups of four students, either one person was invariably left out of the problem solvingprocess or the group split into two pairs. Sometimes the person left out was the more timidstudent who was reticent to ask for clarification. At other times the person left out was themost knowledgeable student who appeared to tire of continually trying to convince the threeother group members to try an approach, and resorted to solving the problem alone. Toverify these observations, we counted the number of contributions each group member madeto a constant-acceleration kinematics problem from the videotapes of a typical three-memberand four-member group. Each member of the group of three made 38%, 36%, and 26% of thecontributions to the solution. For the group of four, each member made 37%, 32%, 23%, and8% of the contributions to the solution. The only contribution of the least involved student(8%) was to check the numerical calculations.Group Size and AssignmentWe found that the optimal group size is three for students not experienced ineffective group work (see Figure 8.1). In retrospect, the reason is almost obvious.Groups of two have no simple mechanism for deciding between two strongly heldopinions. Within a group of three, each of the proponents must explain their idea toa third person. Also a group of two introductory students often lacks some physicsknowledge necessary to attack a problem. With a group of four it is difficult foreach student to contribute to the problem solution in the time they have. Ourresearch indicated that one member is usually quiet, although they may be aneffective group member in well-functioning groups. Of course, if your class is notdivisible by three, then you will have some pairs or four-member groups. We foundthat four-member groups generally work better than pairs. [The exception to this

Chapter 8: Managing Groups and Appropriate Group Problems 99rule-of-thumb is if students are working with computers -- then pairs are preferableto groups of four.]We recommend assigning students to mixedachievementgroups based on past problem-solvingperformance on exams, rather than letting studentsform their own groups (see Figure 8.2). Below are theadvantages of group assignment.Optimal Learning. The most important reason toassign students to groups is because past research incooperative group learning (including our own research) indicates that students learnmore when they work in mixed-performance groups than when they work inhomogeneous-performance groups. We do not, however, want students to label thehigh, medium and lower-performance students in their groups, so we do not tellthem how we assign group membership (see also Chapter 11, page //).Attitude Advantage. It is much easier to set and enforce rules in the beginning ofa class and loosen the enforcement later, than to do the opposite. No matter whatyou do, you will have to change at least some group members to avoid dysfunctionalgroups. If you assign groups at the beginning, you will have fewer disgruntledstudents. You can loosen the rule of assigned groups later in the course if yourstudents get to know each other and become experienced in effective teamwork.This is an example of the 2nd Law of Instruction: Don't change course in midstream;structure early then gradually reduce the structure.Practical Advantage. There are practical reasons for assigning students togroups. For example if you teach in a large commuter school, most students do notknow each other at the beginning of class. They would feel very uncomfortablebeing told simply to "form your own groups." If you teach at a small residentialcollege, students may know each other but have established behavior patterns thatare not based on learning physics, and often not conducive to it. Assigning groupsallows the natural breakup of existing social interaction patterns.Changing GroupsThere are both optimal-learning and practical reasons for changing groups.Avoid Homogeneous Groups. One reason to change groups is that you are likely tostart with many homogeneous-achievement groups, which is not optimal for studentlearning. Normally you do not know the problem-solving performance of yourstudents at the beginning of class. With a small number of students, there can belarge random fluctuations in the achievement-mix of your groups.Avoid Role Patterns. In groups, the necessity to verbalize the procedures, doubts,justifications and explanations helps clarify the thinking of all group members.Students both practice and observe others perform these roles, so they becomebetter individual problem solvers. If students stay in the same group too long, they

100 Part 2: Using Cooperative Problem SolvingFigure 8.2. Why it better to assign students to mixed-achievement groupsIn our research, we examined the written problem solutions of both homogeneous and mixedachievementgroups (based on past problem-solving test performances). The mixedperformancegroups (i.e., a high, medium and lower performing student) consistentlyperformed as well as high performance groups, and better than medium and low performancegroups. For example, our algebra-based class was given a group problem that asked for thelight energy emitted when an electron moves from a larger to a smaller Bohr orbit. 75percent of the mixed-performance groups solved the problem correctly, while only 45% of thehomogeneous groups reached a solution.Observations of group interactions indicated several possible explanations for the betterperformance of heterogeneous groups. For example, on the Bohr-orbit problem thehomogeneous groups of low-and medium-performance students had difficulty identifyingenergy terms consistent with the defined system. They did not appear to have a sufficientreservoir of correct procedural knowledge to get very far on context-rich problems. Most ofthe homogeneous high performance groups included the gravitational potential energy as wellas the electric potential energy in the conservation of energy equation, even though an orderof-magnitudecalculation of the ratio of the electric to gravitational potential energy had beendone in the lectures. These groups tended to make the problem more complicated thannecessary or overlooked the obvious. They were usually able to correct their mistake, butonly after carrying the inefficient or incorrect solution further than necessary. For example,in the heterogeneous (mixed-performance) groups, it was usually the medium or lowerperformance student who pointed out that the gravitational potential energy term was notneeded. ["But remember from lecture, the electric potential energy was lots and lots biggerthan the gravitational potential energy. Can't we leave it out?"] Although the higherperformance student typically supplied the leadership in generating new ideas or approachesto the problem, the low or medium performance student often kept the group on track bypointing out obvious and simple ideas.In heterogeneous groups, the low- or medium-performance student also frequently asked forclarification of the physics concept or procedure under discussion. While explainingor elaborating, the higher-performance student often recognized a mistake, such asoverlooking a contributing variable or making the problem more complicated than necessary.For example, a group was observed while solving a problem in which a car traveling up a hillslides to a stop after the brakes are applied. The problem statement included the coefficientof both static and kinetic friction. It was the higher performance student who first thoughtthat both static and kinetic frictional forces were needed to solve the problem. When thelower-performance student in the group asked for an explanation, the higher-performancestudent started to push her pencil up an inclined notebook to explain what she meant. In theprocess of justifying her position she realized that only the kinetic frictional force wasneeded.tend to fall into role patterns. The result is that they do not rehearse the differentroles they need to perform on individual problems, and consequently do achieveoptimal learning gains.Difficult Students. A third, practical reason forchanging groups is that your first groupassignments may include some dysfunctionalgroups (because of personality conflicts).Students find it miserable to contemplateworking a whole semester with someone whoisn’t compatible, and may disengage. However,

Chapter 8: Managing Groups and Appropriate Group Problems 101most will accept the challenge of working together if they know that it is for alimited time. After you get to know the students better, you can place the "difficult"students in a better group. Strategies for dealing with difficult group members arediscussed in Chapter 13.Individual Responsibility. Finally, one of the most important reasons to changegroups is to reinforce the importance of the individual in cooperative problemsolving. The most difficult point in the course for group management is the firsttime you change groups. By that time most groups have been reasonably successful,and students are convinced they are in a “magic” group. Changing groups elicitsmany complaints, but is necessary for students to learn that success depends onindividual effort and not on a particular group.Frequency of Group Change. Students need to work in the same group longenough to experience some success. The frequency of changing groups candecrease over the course as students become more confident and comfortable withCPS. For example, we change groups about 3 - 4 times in the first semester, butfewer times in the second semester. Since students are very sensitive to grades(Zeroth Law of Instruction), we change groups only after a class exam. Theinformation from that exam also provides useful input for assigning groups (seeChapter 11).Group Role Assignment and RotationThere are many different roles that can beassigned for different types of tasks. Forproblem solving, we assign planning andmonitoring roles that students have toassume when they solve challengingproblems individually -- Manager,Checker/Recorder, andSkeptic/Summarizer. When an expertsolves a problem, they continually organizeand modify a plan of action, making surethey don't loose track of where they areand what they need to do next. These areHmmm. As aSkeptic, what shouldI be noticing?the internal management functions. At the same time, they function as recordercontinually checking their solution to make sure it follows a logical and organizedpath. Finally, the expert is continually skeptical, asking questions about each step --"What other possibilities are there? Should I apply a different principle to solve thisproblem?"Most students do not exhibit these reflective (metacognitive) practices when solvinga problem so the group roles we give them allows them to practice this behavior. Adescription of the group roles we give to students is shown in Figure 8.4. Since theroles emphasize the reflective skills that students need when solving a problemindividually, we do not assign socially useful roles such as leader, facilitator, orconciliator. These roles are assumed naturally by members of the group.

102 Part 2: Using Cooperative Problem SolvingFigure 8.3. Example of definition of group roles.In your discussion section for this course, you will be working in cooperative groups to solvewritten problems. To help you learn the material and work together effectively, each groupmember will be assigned a specific role. Your responsibilities for each role are defined on thechart below.ACTIONSWHAT IT SOUNDS LIKE*MANAGERDIRECT THE SEQUENCE OF STEPS.KEEP YOUR GROUP "ON-TRACK."MAKE SURE EVERYONE IN YOUR GROUPPARTICIPATES.WATCH THE TIME SPENT ON EACH STEP.RECORDER/CHECKERACT AS A SCRIBE FOR YOUR GROUP.CHECK FOR UNDERSTANDING OF ALLMEMBERS.MAKE SURE ALL MEMBERS OF YOUR GROUPAGREE WITH EACH THINK YOU WRITE.MAKE SURE NAMES ARE ON SOLUTION.SKEPTIC/SUMMARIZERHELP YOUR GROUP AVOID COMING TOAGREEMENT TOO QUICKLY.MAKE SURE ALL POSSIBILITIES AREEXPLORED.SUGGEST ALTERNATIVE IDEAS.SUMMARIZE (RESTATE) YOUR GROUP'SDISCUSSION AND CONCLUSIONS.KEEP TRACK OF DIFFERENT POSITIONS OFGROUP MEMBERS AND SUMMARIZE BEFOREDECIDING."First, we need to draw a picture ofthe situation.""Let's come back to this later if wehave time.""Chris, what do you think about thisidea?""We only have 5 minutes left. Let'sfinish the algebraic solution."Do we all understand this diagram Ijust finished?""Explain why you think that . . . .""Are we in agreement on this?""Here, sign the problem we justfinished!""What other possibilities are therefor …?""I'm not sure we're on the righttrack here. Let's try to look at thisanother way. . . .""Why?""What about using . . . . instead of. . . . ?"So here's what we've decided sofar.""Chris thinks we should . . . . , whilePat thinks we should . . . ."

Chapter 8: Managing Groups and Appropriate Group Problems 103Figure 8.4. Research on assigning and rotating group roles.Our observations of group interactions after we assigned roles indicated that the number ofdysfunctional groups (e.g., one student dominates; students cannot resolve a difference ofopinion) at any given time decreased from about 40% (2 in 5 groups) to about 10 - 20% (lessthan 1 out of every 5 groups). With fewer dysfunctional groups, an instructor has more timefor appropriate and timely intervention to coach physics, optimizing the learning of allstudents. Our interviews confirmed that students in groups with assigned and rotated rolesbecame more comfortable with their group’s interactions, particularly at the beginning of thecourse.In well functioning groups, members share the roles of manager, checker, skeptic andsummarizer, and role assumption usually fluctuates over time. As long as thereflective problem solving functions are being displayed and practiced, students donot need to be reminded to "stick to their roles." Unfortunately, most of these rolesare not in evidence, especially at the beginning of the course, so we have to assignthem.We also continue to assign group roles because they are an efficient technique toreduce the number of dysfunctional groups (see Figure 8.4). Students indysfunctional groups are not learning. The roles help reduce the number ofdysfunctional groups in several ways described in Chapter 13.Individual Responsibility. At the beginning of an introductory class, many studentshave never participated in cooperative problem solving and do not know what theyare supposed to do. The roles remind them of appropriate individual actions in agroup.Optimal Learning. Assigning roles allows students to practice behavior that may notbe natural or even socially acceptable. For example, “I don’t want to be bossy, but Iam the manager. Let’s move on to . . ..” In addition, we initially had some studentswho were too polite to disagree openly with the ideas of other group members. Therole of “Skeptic” allowed these students a socially acceptable way to disagree.Coaching Groups. The group roles provide anWho is theManager?efficient and effective way for the instructor tocoach groups that are having difficulty applyingphysics concepts and principles to solve theproblem. These techniques are discussed inChapter 13. Remember the 2nd Law ofInstruction: Don’t change course in midstream; structureearly then gradually reduce the structure. This means itis very difficult to assign roles when you finallydiscover you need them. As students becomemore comfortable and competent with CPS, the group roles slowly and naturally"fade" away from students' minds, except when you intervene with an occasionaldysfunctional group.

104 Part 2: Using Cooperative Problem SolvingFigure 8.5. An example of a Group-Functioning Evaluation SheetDate: Group #:1. Use the following grid to rate yourself on your participation and contributionsin your group’s problem solving. Also, agree on a group rating. 0 = Poor,1 = Fair, 2 = Good, 3 = Excellent.Name Name Name Name Groupa. Participation in solvingproblem.b. Contribution of ideas tothorough analysis of theproblem before generatingappropriate equations.c. Contribution of ideas toplanning a math solution(before the numericalsolution).d. Overall use of a logical,organized approach tosolving the problem.2. What are two specific actions we did today that helped us work togethertowards a successful solution?3. What is a specific action that would help us do even better next time?Group Signatures:Manager: .Skeptic: .Recorder/Checker: .Summarizer: .

Chapter 8: Managing Groups and Appropriate Group Problems 105Group ProcessingOne of the elements that distinguish cooperativegroups from traditional groups is structuringoccasional opportunities for students to discuss howwell they are solving the problems together and howwell they are maintaining effective workingrelationships among members. For this purpose, weuse the Group Functioning Evaluation form shown inFigure 8.5. After the group has discussed andcompleted the evaluation, the instructor spends a few minutes in a class discussionof the answers to Question 6, so students can consider a wider range of ways groupscould function better. Common answers include: "Come better prepared; Listenbetter to what people say; Make better use of our roles (e.g., "Be sure the Managerwatches the time so we can finish the problem." or "Be sure the Skeptic doesn't letus decide too quickly.").In our studies, we found that when students were given a chance to discuss theirgroup's functioning, their attitude about group problem solving improved. Therewas also a sharp decrease in the number of students who visited instructors duringoffice hours to complain about their group assignment. In addition, groups thatwere not functioning well improved their subsequent effectiveness following thesediscussions. For example, in groups with a dominant student, the other groupmembers were more willing to say things like: Hey, remember what we said lastweek. Listen to Kerry. She's trying to explain why we don't need all thisinformation about the Lunar Lander's descent." In groups that suffered fromconflict avoidance, there were comments like: "Oops! I forgot to be the skeptic.Let's see. Are we sure friction is in this direction. I mean, how do we know it's notin the opposite direction?" As usual, this result was consistent with the research onpre-college students. 2Appropriate Group ProblemsWith appropriate grading, course structure, and group management in place, wefound that even the more difficult, end-of-chapter textbook problems were usuallynot effective learning tools when used for either individual exams or for groupproblems (see Chapter 3). For example, one semester we gave students all therelevant equations from their textbook and the group problem shown in Figure 8.6.Although this problem has a minimal context, group discussions tended to revolvearound “what formulas should we use,” rather than what physics concepts should beapplied to this problem.” An illustration of a typical group solution for this problemis shown In Figure 8.6. The students in this group did not begin with a discussionand analysis of the forces acting on the carton in this situation. Instead, theyattempted to recall the force diagram from their text, which were for a block slidingdown an inclined plane. Consequently, their solution has the frictional force in thewrong direction and the force equation has a sign error. The students began by

106 Part 2: Using Cooperative Problem SolvingFigure 8.6. A Typical Group Solution to a Textbook Problem. The arrows have been added toshow the progression of the mathematical solution. 3Textbook Problem. A 5.0-kg carton slides 0.5 m up a ramp to a stop. The ramp is at anangle of 20o to the ground, and the coefficient of kinetic friction between the cartonand the ramp surface is 0.60. What is the initial velocity of the carton?Nf=NN mg cos fmg= 0.6 Nf = 0.6 46= 27.6=5 9.80.93=46f - mg sin =ma27.6 - (5 9.8 0.34) = 5a27.6 - 16.8 a52.17 av at 2.17 0.48v = 1.04 m/sv at 2.17 t0.5 2.17 tt0.52.17 t20.48 tx vt0.5 vt0.5 vthaphazardly plugging numbers into formulas until they had calculated a numericalanswer. Their conversation concerned finding additional formulas that containedthe same symbols as the unknown variables: “Can’t we use this distance formula[x = vt]? It has v and t in it.” They did not discuss the meaning of the symbols orformulas, and they incorrectly combined an equation containing an instantaneousvelocity (v = at) with a one containing an average velocity (x = v ave t) to calculate theinitial velocity of the block. From observations, interview data, and the examinationof group problem solutions, we estimated that about two-thirds of the groups usedthis “formulaic” problem-solving framework rather then the logical, organizedproblem-solving framework modeled during lecture. This solution is a combinationof novice strategies described in Chapter 2 (pages // - //), pattern matching andplug-and-chug.

Chapter 8: Managing Groups and Appropriate Group Problems 107Figure 8.7. The Traffic Ticket problemYou are driving up a steep hill when suddenly a small boy runs out in the street. You slam onyour brakes and skid to a stop, leaving a 50-foot skid mark on the street. As you are recoveringfrom the shock, a watching policeman comes over and gives you a ticket for exceeding the 25mph speed limit.You wonder whether you might fight the traffic ticket in court, so before you leave thescene you collect some information. You determine that the street makes an angle of 20with the horizontal. The coefficient of static friction between your tires and the street is0.80, and the coefficient of kinetic friction is 0.60. Your car’s information book tells youthat the mass of your car is 1570 kg. You weigh 130 lbs, and a witness tells you that theboy had a weight of 60 lbs and took about 3 seconds to cross the 15-foot wide street.Should you fight the traffic ticket in court?Effective group problems follow the 3rd Law of Instruction: Make it easier for studentsto do what you want them to do and more difficult to do what you don’t want. That is, groupproblems should be complex enough so there is a real advantage to discussing boththe problem situation and applicable physics before plowing ahead. We will use thetraffic-ticket problem in Figure 8.7, which is a modification of the textbook problemin Figure 8.6, to illustrate some of the criteria for a good group problem.Criteria 1. A group problem must be designed so that: There is something to discuss initially so that everyone (even the weakestmember) can contribute to the discussion. For example in the traffic-ticketproblem, students spend time initially determining how the car is moving anddrawing a picture of the situation. Early successful contributions encouragestudents to make larger contributions later. There are several decisions to make in solving the problem. For example inthe traffic-ticket problem, students have to decide what quantity to calculateand which information is relevant to the solution.Criteria 2. A group problem must be challenging enough so that: Even the best student in the group cannot immediately see how to solve theproblem. Knowledge of basic physics concepts isnecessary to interpret the problem. Thetraffic-ticket problem discourages thealgorithmic use of formulas in severalways. For example, students must decidewhether the 3-second time the boy tookto cross the street is relevant to findingthe initial velocity of the car just beforethe brakes were applied. Students' physics difficulties arise naturally and must be discussed. Forexample, many students confuse static friction and kinetic friction. To solve

108 Part 2: Using Cooperative Problem Solvingthe traffic-ticket problem students must discuss the meaning of each, andwhich type of friction should be applied in this situation. Students feel good about their role in arriving at the solution. There areenough decisions to discuss in the traffic-ticket problem that all studentscontribute, even the quiet students. We found in our research that it is oftenthe quiet student who requests an explanation or clarification that moves thegroup forward. 4Criteria 3. At the same time, the group problem must be simple enough so that: The mathematics is not excessive or complex. The solution path, once arrived at, can beunderstood, appreciated, and easilyexplained to all members of the group.For example, the traffic-ticket problemcan be solved by the straight-forwardapplication of kinematics concepts andNewton’s Second Law. A majority ofgroups can reach a solution in the timeallotted.The traffic-ticket problem has many traits that make it a challenging problem, evenfor a group. Striking the right balance between complexity and simplicity forproblems is difficult for the instructor. We have developed and tested a set ofcriteria for judging the difficulty of a problem to decide if they would be suitable forgroup practice or exams (see Chapter 16). It is probably not an accident that thecriteria for designing a problem that encourages learning physics, given in Chapter 3,also results in a useful group problem.Endnotes1 Heller, P. & Hollabaugh, M. (1992). Teaching problem solving throughcooperative grouping. Part 2: Designing problems and structuring groups,American Journal of Physics, 60(7), 637-644.2 See Johnson, D. W. and Johnson, R. T. (1989), Cooperation and competition: Theoryand research, Edina MN, Interaction; and Johnson, D.W., Johnson, R.T., andSmith, K.A. (2007), The state of cooperative learning in postsecondary andprofessional settings. Educational Psychology Review, 19 (1), 15-293 Heller, P., Keith, R., & Anderson, S. (1992). Teaching problem solving throughcooperative grouping. Part 1: Groups versus individual problem solving, AmericanJournal of Physics, 60(7), 627-636.4 Hollabaugh, M. (1995), Physics problem-solving in cooperative learning groups, Ph.D. Thesis,University of Minnesota

Chapter 9Course Structure and Grading forCooperative Problem Solving109In this chapter: Scheduling for continuity of group interactions and finding adequate assistance and rooms forCPS. What is the appropriate grading in a CPS course. How to give consistent grades and feedback for students’ problem solutions. How to get started with informal groups in lecture.The learning advantage ofcooperative problem solving(CPS) sessions lies in thestudents’ co-construction of a problemsolution. There are several aspectsof structuring a course forcooperative problem solving thatmake it easier for students to coconstructgroup solutions than tosolve the group problemsindividually (3rd Law ofInstruction). These structuresinclude scheduling cooperative group work, adequate assistance, and adequaterooms. In addition, effective implementation of CPS sessions requires appropriatecourse grading, including overall grading for the course and consistent grading andfeedback of students’ problem solutions.These structures are described in this chapter. The last section contains suggestionsfor how to get started with informal groups in your lecture.Remember, the success of CPS depends on effective demonstrations of the use of aresearch-based, problems-solving framework to solve problems during your classes.

110 Part 2: Using Cooperative Problem SobvingFigure 9.1. Outline for demonstrating the use of a research-basedproblem-solving framework when solving problems in class1. Adapt a research-based problem-solving framework that emphasizes the application offundamental physics principles (e.g., kinematics, Newton’s second law, conservation of energy) tosolve problems. A problem-solving framework is a logical and organized guide to arrive at aproblem solution. It gets students started, guides them to what to consider next, organizes theirmathematics, and helps them determine if their answer is correct. [See Chapter 14 for how topersonalize a framework for the needs of your students.]2. Demonstrate how to use the framework every time you solve a problem in class.* Show every step, no matter how small, to arrive at a solution. Explain all decisions necessary to solve the problem. Always use the same framework, no matter what the topic. Hand out or have on your website examples of complete solutions to problems emphasizingthe physics decisions and showing every step. After some time, allow each student to make their own reasonable variations of theframework for their solutions.3. Demonstrating the problem solving process can actively engage students. Always allow the students several minutes to read the problem and begin their solution beforemodeling begins. Pause several times while demonstrating the problem solving framework to ask students asimple question and allow several minutes for students to:• write their answer or answer electronically;• turn to their neighbor and discuss the answers to the question; or• work in an informal group of 3 to answer the question. The question may be an elaboration of a step in the problem-solving process, a simpleapplication of the point just demonstrated, or the next step in the solution process.* It is difficult to put yourself in the minds of students and demonstrate the use of a problem-solvingframework to solve a problem. Remember, the “problems” in introductory physics are not realproblems for you. It is difficult to ask yourself continually: “What would I do next if I didn’t knowhow to solve this problem already?”Scheduling: Continuity of Group InteractionsTo successfully co-construct a problem solution,students need to have time to discuss the physicsand a procedure to solve the problem, and to becoached in problem solving by an instructor. Thistypically requires at least 30 minutes of continuousworking time. Taken together with the brief introduction to the problem and asummary discussion, this means one group problem session takes about 50 minutes.We found that to make a substantial impact on student learning, students also needto solve at least two problems per week while being coached in groups. The reasonis mostly a matter of building trust in a group. If students work as a group onlyonce a week, it takes them longer to get to know each other enough to establishtrust, so they feel comfortable sharing their conceptual and procedural knowledge

Chapter 9: Course Structure for Cooperative Problem Solving 111and requesting explanations and justifications. You can, however, achieverespectable improvement in students’ performance when students solve one groupproblem each week if students work in the same groups in other aspects of yourcourse, such as their lab.If you have block scheduling (1.5 – 2 hours) or a studio format (combined lecture,recitation, and lab), you probably do not have a scheduling problem. If you do havea scheduling problem, there are many creative solutions. For example, we know oneinstructor (who teaches both the lecture and labs) who decided that students did notuse their three-hour lab time effectively – they usually came late and finished early.So the lab time was broken into one hour of CPS and two hours to complete the(same) lab experiments.At our very large university we had more severescheduling problems. Many years ago we startedwith different courses for lecture and the lab, and no“recitation” sections. First we changed to onecourse that included the lab, then we changed thecourse registration procedure and included adiscussion section. Now for each lecture section ofthe course, students also register for one section thatmeets at two scheduled times – a 50 minute “discussion” (CPS) and a two-hourlaboratory. The same instructor coaches the same groups in both the discussionsection and the laboratory. To better use the laboratory time to support learningphysics through problem solving, we developed context-rich laboratory problems. 1What Resources Are Needed to Implement CPS?Two additional and related resources (besides your own time) may be needed toimplement CPS effectively: Adequate assistance (personnel); and Adequate rooms.There are many tradeoffs between these resources, which are both in short supplyat many institutions. Some of the tradeoffs are outlined in the sections below.Adequate AssistanceThe limiting factor in implementing CPS is the numberof students per instructor. It is difficult for anyinstructor to teach 9 or more groups withoutassistance. Experienced instructors can teachcomfortably 7 - 8 groups per class section.Inexperienced (first-year) graduate or undergraduateTAs, however, can handle only 5 groups per section.

112 Part 2: Using Cooperative Problem SobvingIf you are the only instructor of small classes, then no additional assistance is neededto implement CPS. If you are in this fortunate situation, you could skip the rest ofthis section. But if you do not have one experienced instructor per 20-28 students(or one first-year TA for every 15 – 18 students), people have found creative ways tofind free assistance for CPS."Free" Undergraduate Assistance. We know one instructor who teaches anintroductory physics course with 40 students at a small, liberal-arts college. Henegotiated with the Education Department to offer course credit to physicsundergraduate students in their secondary teacher-education program. Theseselected students assist the instructor during the course time set aside for CPS.Change Responsibilities of Paid Assistants. If you have undergraduate orgraduate students who spend all or part of their time grading homework problemsand/or lab reports, consider reassigning all or some of this time for assistance withCPS. For example, we cut down slightly on the number of required lab reports, anddevised a rubric for grading that allows TAs to do some lab evaluation during labtime. We also eliminated or sharply curtailed the grading of homework andredirected this effort to coaching in CPS sections.Other colleges and universities have instituted computerized homework “grading”,or give short multiple-choice tests on the computer each week. Both options allowyou to check whether students can use appropriate mathematics in simple one-ortwo-step exercises. Of course, you can continue to assign some of the moredifficult, end-of-chapter problems for homework, even when they are not graded.To get students to do homework, it is useful to make one problem on your quiz veryclose to a homework problem and tell students you will do so. After the quiz, pointout which problem was very close to the homework problem. Because subtlety islost on most students, it is useful to use similar objects and situations as thehomework problem simply solving for a different variable. The most usefulhomework procedure that we have found is to post a sample quiz on your web siteabout 2 weeks before your next quiz. Do not post the answers or solutions until afew days before the quiz. Many more students will seriously work on problems andget help from the TAs in the tutorial room if you call something a sample quizrather than homework.What Kind of Room is Needed?The ideal room for CPS (in most disciplines, not just physics) is a carpeted roomwith two walls of boards and small, round cocktail-style tables with moveable chairsthat accommodate 3 to 4 students (see Chapter 8, page // for optimal group size).The room has adequate space between groups for the instructor to circulate easilyamong the groups. However, such rooms are rarely available. The minimum roomrequirements for CPS are:1. The room must have sufficient wall space for one person from each group towrite or post simultaneously parts of their group’s problem solution.

Chapter 9: Course Structure for Cooperative Problem Solving 113Figure 9.2. Top diagram view of movable chairs (black circles) in room arranged for a CPSsession. Dashed arrows indicate spaces for instructor to circulate to each group.2. Students must be able to sit facing each other. In other words, for effectivegroup interactions, students need to be able to look directly at each other, kneeto-knee(see Chapter 7, page //).3. The room must be large enough so there is space between groups to allow groupsto function independently and the instructor to circulate to each group.No Boards (or not enough boards). The problem of enough board space can besolved in many ways. If you have storage space in the room, you can make “whiteboards” on which groups can write their diagrams and equations with dry-erasepens. White boards are actually preferable to writing on the blackboard becauseeach group can decide what they want to put on the board together, rather thansending one person to write on the blackboard. Large sheets of whiteboard can bepurchased at hardware/lumber yards and cut into 2 ft. x 3 ft. pieces.If you have neither storage space for white boards nor sufficient board space, youcan use sheets of white butcher paper for groups to write on. The disadvantage ofbutcher paper is the need for a large flat surface to write on (our students sometimesuse the floor), and the necessity to tape the sheet to the wall and remove them at theend of class.The minimum requirements for CPS eliminate rooms with stadium seating and striptables with fixed seats. However, the following rooms meet the criteria.

114 Part 2: Using Cooperative Problem SobvingFigure 9.3. Top diagram view of room with strip tables (rectangles) and movable chairs (blackcircles). Dashed arrows indicate spaces for instructor to circulate to each group.Figure 9.4. Diagram of studio format room at Massachusetts Institute of Technology (MIT) 2

Chapter 9: Course Structure for Cooperative Problem Solving 115No Tables, Moveable Chairs. Minimally you need a room with moveable chairs that islarge enough to accommodate your class size. The room does not have to be in thescience or physics building. We schedule CPS discussion sections in smallclassrooms all over campus. The map in Figure 9.2 is an example how chairs can bearranged so the groups can function independently and the instructor can circulateto each group.Strip Tables with Moveable Chairs. We do not recommend that you implement CPSin a room with rows of long, narrow tables fixed to the floor. But it can be done ifthe room is large enough and the chairs are moveable (or at least swivel). Forexample, the map on Figure 9.3 shows how 8 groups could be arranged in a roomwith eight strip tables. The groups are far enough apart, and the instructor can getto each group.Lab Room. Lab rooms are not ideal, but if you have a small class, you can docooperative problem solving in most lab rooms. You must, however, be able toarrange the groups so students can be sitting facing each other, for example at theend of tables. Of course, all the apparatus must be cleared out of the way.Room with Large Tables. Some universities have replaced stadium seating in largelecture rooms with the studio format – large rooms with tables that seat 6 to 10students, as illustrated in Figure 9.4. There is usually sufficient space for the grouprecorder, in the middle chair, to move his/her chair back away from the table. Theother two group members on either side of the recorder can turn their chairs aroundto face the recorder.Appropriate Course GradingRemember the Zeroth Law of Instruction: If you don’t grade for it, students won’t learnit! Like it or not, grading is the single most important teaching action. This sectiondescribes some recommendations for CPS courses.Absolute Grading CriteriaThe minimal requirement is that your course grades are not be based on a curve -- a“you win, I loose” grading policy. For groups to work, students must know thatwhen they help others, they are not reducing their chances of getting a good coursegrade. To establish absolute grading criteria, you must be able to specify coursestandards in a manner independent of a specific problem. All grading practiceswithin the course must conform to those standards. Initially you can base yourgrading criteria on results from previous classes, so they represent realistic goals ofstudent performance. To guard against an occasional test that is too hard or toolong, one can normalize each exam’s score to a criterion that does not depend onthe performance of the entire class. We typically normalize to a score that ishumanly possible as determined by the highest score on an exam.

116 Part 2: Using Cooperative Problem SobvingNumber of Questions on a TestOne corollary of the Zeroth Law of Instruction is that students ignore (andresent) an activity that they do not see as having a direct effect on theirgrade. In other words, students need an explicit match between the timespent on an activity and how they are graded. For most students to becomebetter problem solvers, as defined in Chapter 4, they need time todemonstrate their learning on tests. This determines the maximum numberof questions on a test. We have found that typical students need about 20minutes to solve a context-rich problem on a test. This time factor makessome students feel rushed, but more time does not help the majority ofstudents arrive at a better solution. 3 This limits 50-minute tests to twoproblems and some multiple-choice questions. To give adequate feedbackto students, we give a test about once every three weeks.Information Available on Tests.For students to see a match between group work and how they are graded,individual test problems should be graded using the same criteria as group problems.Group problems, whether for practice or graded, should always look the same asindividual test problems from the students’ point of view.To discourage memorization of formulas and problem specific algorithms, the basicinformation needed to solve problems, including constants and equations, should besupplied in the same way for the individual tests and all group problems (see pageChapter 3, pages //-//). The information sheet that we give to students is the samefor the group and the individual part of the test.Grading Group Solutions.Another consequence of the Zeroth Law ofInstruction is that if you want students to workeffectively in cooperative groups, then youmust, at least occasionally, grade the groupproduct. That is, some small but significantpart of student's total course grade should befor group problem solving. There are, ofcourse, many ways to do this. You could, forexample, assign 10 - 15% of each student's finalgrade to a fixed number of group problemsolutions. That is, groups occasionally turn inone problem solution for grading, and each group member gets the same grade forthe group solution. We have found that grading every group problem is counterproductive. A constant grade pressure seems to inhibit group discussions that alloweach student to explore their ideas about the physics. Under those conditions,student activity becomes exclusively directed to getting an answer.

Chapter 9: Course Structure for Cooperative Problem Solving 117When a group problem is graded, we have found that integrating it into a regular testsituation makes it more meaningful to our students. For example, each of our testshas a group part and an individual part. The first part of the test is a group problemthat students complete in their discussion sections. The following day studentscomplete the individual part of the test, consisting of two problems and about tenmultiple-choice questions, in the lecture room. No student believes that oneproblem out of three on a test is unimportant. When the final exam, which isentirely individual, is added, the group problems account for about 15% of theirtotal test scores. Other parts of the grade (e.g., individual laboratory reports) reducegroup work to about 10% of the students' course grade.An advantage of this presentation of grading lies in the way students interpret theirtest scores. When groups are well managed (see Chapter 8), the highest score thatstudents receive on a test is almost always for the group problem, which is also themost difficult problem on the test.3 This reinforces the advantages of cooperativegroupproblem solving.Consistent Grading of Problem SolutionsIf your goal is to teach physics through problem solving, then the grading of bothindividual and group problem solutions should reflect the behavior you value inproblem solving. For example, if you value a careful description and analysis of aproblem before the mathematical manipulation of formulas, you should not gradeonly for the appearance of equations leading to a correct answer. Remember theZeroth Law of Instruction: If you don’t grade for it, students don’t do it. From a student’sperspective, grading only for the equations leading to a correct answer requires themto do what you do not want them to do -- solve problems disconnected fromphysics by manipulating formulas or trying to match a memorized mathematicalsolution pattern (see Chapter 3, pages //-//).The consistent grading of problem solutions can provide your students with rewardsfor learning physics through problem solving, and place barriers to novice plug-andchugand pattern-matching approaches. Below are some recommendations for thistype of grading.Grading Based on Problem-solving FrameworkAdopt a grading scheme based on thesuccessful completion of each step of theproblem solving strategy you teach. Forexample, suppose each problem solution isgraded on a 25-point scale. The table on thenext page shows the distribution of points forthe successful completion of each step in theCompetent Problem-solving Strategy for analgebra-based course.

118 Part 2: Using Cooperative Problem SobvingFigure 9.5. Points for grading written problem solutions on different testsSteps First 2 Tests 3 rd Test 4 th TestFocus on the problem 7 5 5Describe the Physics 8 7 6Plan a Solution 5 7 6Execute the Plan 3 4 5Evaluate the Solution 2 2 3TOTAL POINTS 25 25 25This grading scheme is a formidable barrier for students who use the plug-and-chugor pattern-matching technique. For example, the solutions shown in Chapter 2,pages // and // would both get a low grade! On the first test, this is a big shock tothose students who are accustomed to getting partial credit for calculatingsomething, even if it makes no sense within physics.Drop the Lowest Test Score.In the first semester we give a test every 3 weeks,but drop the lowest test score for calculating thefinal course grade. This allows students to dopoorly on the first test, and still get a good coursegrade -- if they start using a logical and organizedproblem-solving strategy. Unfortunately, thispractice has a consequence that is detrimental tostudents who do well on all of the tests. Theysee no point in taking the last test in the coursesince they will drop one of their good scores.Following the Zeroth Law of Instruction, theydon’t attend to that material and do poorly onthat part of the final exam. To avoid thisbehavior, our grading scheme gives a lowerweight to the final exam if no test score is dropped. For the student, it is alwaysworth doing well on a test. For the same reason, no student is allowed tocompletely drop the final examination.Change the Point Distribution for Grading Problem Solutions.Periodically changing the number of points allocated to each step in your problemsolvingstrategy (see Figure 9.5) can provide students with a ladder up the learningmountain (see Chapter 1, page //). For example, during the first weeks of thecourse, many of our students have difficulty carrying through an entire problemsolution. To emphasize the importance of the initial qualitative steps in problem

Chapter 9: Course Structure for Cooperative Problem Solving 119Figure 9.6. Grading Feedback for Students' Problem SolutionsFOCUS ON THE PROBLEM AND DESCRIBE THE PHYSICS1. Picture or Diagram is misleading or inaccuratea. missing important objects or interactionsb. includes spurious objects or interactionsc. other incorrect diagrammatic translations of problem information2. Relevant variables not assigned and clearly labeleda. many important variables not definedb. defined variables not clearly distinguished from each other3. Approach invalid, too vague, or missinga. application of principles inappropriateb. misunderstanding of fundamental principlec. simplifying approximations not stated or inappropriate4. Necessary fundamental principles missing5. Incorrect or invalid statement of known values or assumptions6. Incorrect assertion of general relationships between variablesa. application of principles to inappropriate parts of the problemb. incorrectly assumed relationship between unknown variables, such as T1=T2.c. overlooked important relationship between variables, such as a1=a2.d. misunderstanding of fundamental principle7. Incorrect statement of target variable or no target stateda. target variable doesn't correspond to question in Approachb. does not explicitly state target variablec. wrong target8. Major misconceptionPlan the Solution9. Poor use of the physics description to generate a plana. physics description was not used to generate a planb. inappropriate equation(s) was introducedc. undefined variables used in equations10. Improper construction of specific equationsa. inappropriate substitution of variables into general equationsb. numerical values were substituted too soon11. Solution order is missing or uncleara. there is no clear logical progression through the problemb. solution order can't be understood from what is written12. Plan can not be executeda. there are not enough equationsb. a relationship was counted more than onceExecute the Plan and Evaluate the Solution13. Execution is illogicala. Incorrect physics was introduced to solve the problemb. unacceptable mathematical assumption was used14. Mistake in executiona. algebra mistakeb. used incorrect values for known variablesc. used “math magic” (e.g., m 1=m 2=1, took square root of negative number)15. Did not check units and/or evaluate answer

120 Part 2: Using Cooperative Problem Sobvingsolving, we give students the most points for these steps in the first weeks of thecourse. As the course progresses and students become more knowledgeable andcomfortable with all the problem-solving steps, the allocation of points is moreevenly distributed, as illustrated in Figure 9.5.Feedback on Students’ Problem SolutionsAlthough it is helpful to give students extensive comments on each problemsolution, such feedback is excessively time consuming for a large class. Wedeveloped a chart of common student mistakes, shown in Figure 9.6, to help gradeproblems consistently and to give feedback. If the chart is available to students,then as mistakes are encountered on problem solutions, the grader can indicate themby number (e.g., 2b, 5, 7a).Getting StartedTo adopt CPS, it is helpful to be in the frame of mindof an experimenter -- cut yourself some slack andtake the long-term view. Implementing a newteaching technique is similar to implementing acompletely new measurement technique in alaboratory. When you start, it naturally takes moretime and effort than the old, comfortable technique.More frustrating still, the first time you "turn it on," iteither doesn't work at all, or you don't get theexpected results. You have to tinker with it, make adjustments, and fine-tune thetechnique until you get the optimal results.If you and your students have no experience with active learning techniques, werecommend that you start with informal groups in the lecture before you implementCPS. Informal groups involve asking a short question during a lecture, having yourstudents turn to their neighbor(s) to discuss the answer and come to consensus.This time-honored technique for improving lectures has been given many names,such as the “one-minute paper,” “peer instruction,” as well as “informal”cooperative grouping.” Some example questions for two-dimensional motion areshown on the following pages. After students discuss the answer for a few minutes,the instructor asks for a call of hands for each answer (or use an electronic system ofclickers).There are many times during a lecture you can stop and ask a short question ofinformal groups. For example, you can ask a question before you start a lecture tofind out what students already know, and focus their thoughts on the lecture tocome (see Figure 9.7a). You can ask a question after lecturing for some time to seeif students have understood the main ideas of your lecture (see Figure 9.7b). Whenyou are demonstrating how to solve a problem, you can ask students about the nextstep in the problem solution (see Figure 9.7c).

Chapter 9: Course Structure for Cooperative Problem Solving 121Figure 9.7. Examples of group questions to use in lecture.Figure 9.7a. Example of group question to use before a new topic is introduced.BCD1. A ball is thrown into the air and follows the path shown at left.AE1 2 3 4 5 6Which arrow best represents the direction of the ball’s acceleration at point B?At point C? At point D?Which arrow best represents the direction of the ball’s velocity at point B?At point C? At point D?Figure 9.7b. Example of a group question to check for understanding after the topic is introduced.Three identical balls are simultaneously thrown with the threeV 1velocities shown by the vectors in the diagram at right.Ignoring air resistance, which of the following statements isV 2true?A. They all move through the air with the same speed.V3B. They hit the ground at the same place.horizontalC. They remain in flight for the same length of time.D. While all three balls are in flight, they are always at the same vertical distance abovethe ground.E. While all three balls are in flight, they are always the same horizontal distance from thestarting point.Figure 9.7c. Example of group question to use for moving on to the next step..A ball rolls off a flat roof that is 5 meters high. One second later, the ball lands 15 metersfrom the house at an angle from the ground. When the ball lands, the horizontalcomponent of its velocity (vx) is 15 meters/second and the vertical component of its velocity(vy) is 10 meters/second.5 mV x = 15 m/sV y = 10 m/s15 m10 m/sA. tan =15 m/sB. tan =15 m/s10 m/sC. tan = 5 m15 mD. tan = 15 m5 mE. tan =10 m/s5 m

122 Part 2: Using Cooperative Problem SobvingFigure 9.8. Examples of informal group questions for before and after a demonstration.Figure 9.8a. Example prediction question before a demonstrationTwo balls will be released from the same height at the same time. Ball A is dropped straightdown, while B is given a horizontal kick. Which ball do you think will hit the floor first?A B A BA. Ball A will hit the floor first.B. Ball B will hit the floor first.C. Both balls will hit the floor at the same time.D. There is not enough information is given.E. I don’t have any idea.Figure 9.8b. Example confirmation question after a demonstration.Two balls were released from the same height at the same time. Ball A was dropped straightdown while ball B was given a horizontal kick. Which ball hit the floor first?A. Ball A hit the floor first.B. Ball B hit the floor first.C. Both balls will hit the floor at the same time.D. I didn’t hear when the balls hit the floor.Finally, it is very helpful to ask questions before and after a demonstration. Aprediction of what students think will happen in the demonstration (and why) helpsfocus student’s attention on the purpose of the demonstration and provides youwith information about your students’ alternative conceptions (misconceptions). Anexample prediction question before a demonstration is shown in Figure 9.8a.Because some students’ alternative conceptions are so strong that they “see” whatthey expect to see, also ask students what they observed right after thedemonstration (see Figure 9.8b).Several resources are available for conceptual questions that can be used forinformal groups during lectures. In his book Peer Instruction: A User’s Manual, EricMazur 4 provides many conceptual questions. There are many good questions inLillian McDermott and Peter Schaffer’s book, Tutorials in Introductory Physics andHomework Package. 5 In addition, Tom O'Kuma, David Maloney, and CurtisHieggelke 6 have published ranking tasks, which make excellent conceptual questionsfor informal groups, in their book Ranking Task Exercises in Physics: Student Edition.

Chapter 9: Course Structure for Cooperative Problem Solving 123There are other advantages to starting slowly with informal groups in your lecture.First, it gives you a baseline from which to judge whether CPS is successful for you.Before you start CPS, you can collect some data from your course. For example,you could collect student answers to informal-group questions. You could also givethe Force Concept Inventory 7 to your students as a pretest and posttest so you cancompare your students’ gains with national norms for this test at similarinstitutions. 8 You can also examine a sample of problem solutions from each test todetermine the kinds of errors are your students making and how well they areexpressing themselves.Second, you may need time to review the available problem-solving frameworks, andmodify these frameworks to match your preferences (see Chapter 14). It is behelpful to practice demonstrating your preferred framework when you solveproblems in lecture. We have found professors often have difficulty puttingthemselves in the minds of students and demonstrating a competent framework.Remember, the “problems” in introductory physics are not real problems for you. Itis difficult to ask yourself continually: “What would I do next if I didn’t know howto solve this problem already?”Remember the 2nd Law of Instruction. Don’t change course in midstream. We do notrecommend starting CPS in the middle of a course. The students have already settheir behavior patterns and will resist any changes.ENDNOTES1For examples of problem-solving labs, go to our website:http://groups.physics.umn.edu/physed/Research/PSL/pslintro.html.2For more information about Technology Enhanced Active Learning (TEAL)classrooms ant MIT, go to http://web.mit.edu/edtech/casestudies/teal.html.3 For the data supporting this statement, see Heller, P., Keith, R., & Anderson, S.(1992). Teaching problem solving through cooperative grouping. Part 1: Groupsversus individual problem solving, American Journal of Physics, 60(7), 627-636.4Mazur, E. (1992). Peer instruction: A user's manual, Upper Sadler River NJ, PrenticeHall.5McDermott, L.C. and Shaffer, P.S., (2002). Tutorials in introductory physics andhomework package, Upper Sadler River NJ; Prentice Hall6O’Kuma, T.L., Maloney, P.D., & Hieggelke, C.J. (2000). Ranking task exercises inphysics: Student edition, Upper Sadler River NJ, Prentice Hall7Hestenes, D., Wells, M., & Swackhamer, G. (1992). Force concept inventory, ThePhysics Teacher, 30, 141-158

124 Part 2: Using Cooperative Problem Sobving8Hake, R. (1998). Interactive-engagement vs traditional methods: A six thousandstudent survey of mechanics test data for introductory physics courses. AmericanJournal of Physics, 66: 64-74.

125Chapter 10Results for Partial and Best-practiceImplementation of CPSIn this chapter: Research investigations of students’ improvement problem-solving skills with CPS Research investigations of students’ improvement conceptual understanding of physics asmeasured by a multiple-choice test (FCI) and open-ended written questions Research investigations of the effect of partial and full implementation of CPS on students’conceptual understanding Research investigations of students’ improvement in learning attitudes with CPSPart 1 provided background about the unsuccessful problem-solving strategiesof beginning students, how context-rich problems and a problem-solvingframework help students engage in real problem solving, and why the bestpracticeimplementation of Cooperative Problem Solving (CPS) is a useful tool forteaching physics through problem solving. Chapters 7 – 9 describe the foundationsof Cooperative Problem Solving and provide detailed information about how toimplement CPS for maximum effectiveness.In this chapter we describe research resultssupporting the use of CPS to improve bothstudents’ problems-solving skills and theirconceptual understanding of physics. A few resultswere published previously, 1 and many results werepresented as contributed papers and AmericanAssociation of Physics Teachers (AAPT) meetings.In this book, we have focused on the results. Whennecessary, comments about methodology appear inthe endnotes.Improvement in Problem-solving PerformanceThe best-practice implementation of Cooperative Problem Solving (CPS) includesteaching students an explicit problem-solving framework and having studentspractice implementing the framework by solving context-rich problems incooperative groups. The full model follows all of the recommendations in thisbook, as outlined below.

126 Part 2: UsingCooperative Problem Solving1. Model (demonstrate) solving problems in lecture using of a research-basedproblem-solving framework which emphasizing the basic principles of physics(Chapters 4 and 14) and establishing a culture of expert practice (Chapter 6).2. Use the scaffolding recommended in this book: worksheets with cues from theframework (Chapter 5); problem solutions on the worksheets (Chapter 5); andinformation sheets (Chapters 3 and 11). Students are required to solveproblems on the worksheets for the first few exams.3. Require students to buy a resource booklet that includes: An explanation of each step of the problem-solving framework. A one-page outline of the framework. Flow-charts for each step in the framework. A blank worksheet with cures from the framework (Figure 5.3). Studentscan photocopy the worksheet for practice. Three sections (kinematics, dynamics using Newton’s second law, and theconservation of energy). Each section includes an explanation of how todraw the appropriate physics diagrams (motion diagrams, force diagrams,and energy tables), several problems with problem solutions on theworksheets, and context-rich problems for students to practice solvingusing the problem-solving framework.4. Work more closely with the teaching assistants to write appropriate contextrichproblems (Chapters 3, 11, and 15) and help them with diagnosing,monitoring, and intervening with groups (Chapter 13).5. Adopt grading practices that require students to use the problem-solvingframework and get the most out of their CPS sessions (Chapter 9)6. Gradually fade this structure over the course of the first semester:The original research in the improvement in problem solving performance in bestpracticeCPS classes was done in the early 1990’s with the algebra-basedintroductory course at the University of Minnesota. A change in students’ problemsolvingperformance over time is very difficult to measure because conceptualunderstanding of the physics is a necessary but not sufficient condition for goodproblem solving performance. 2 In other words, a competent problem solver cannotsolve a problem correctly if their understanding of the physics is imperfect.Consequently, we developed and validated scales for six expert-like problem-solvingskills for rating students’ written problem solutions.• General Approach: Does the physics description in the solution reveal a clearunderstanding of physics concepts and relations?• Usefulness of Description. Does the physics description include the essentialknowledge necessary for a solution• Specific Application of the Physics: Starting from the physics they used, how welldid the student apply this knowledge to the problem situation?• Reasonable plan. Does the solution indicate that sufficient equations wereassembled before the algebraic manipulation of equations was undertaken.

Chapter 10: Results for Partial and Full Implementation of CPS 127Figure 10.1a. Example of a student’s problem solution at the beginning of anintroductory course, showing a lack of logical progression in the solution.Figure 10.1b. Example of a student’s problem solution showing improved logical progressionnear the end of an introductory course in which students are taught an explicit problem-solvingframework and practice implementing the framework by solving context-rich problems incooperative groups.

128 Part 2: UsingCooperative Problem SolvingFigure 10.2. Percentage of the top third, middle third, and lower third of the class whosesolutions followed a logical progression. The dashed lines are included for easeof reading the graphs.PercentStudents10090807060504030201000 20 40 60 80 100 120 140 160Time (days)Top ThirdMiddle ThirdBottom third• Logical Progression: Is the solution logically presented? (see Figure 10.1• Appropriate Mathematics: Is the math correct and useful?The scales for the last three skills are based on the physics that students used tosolve the problem, regardless of whether the approach was correct, incomplete, orincorrect. The ratings for the six skills were equally weighted and normalized toyield an ordinal problem solving scale with a maximum score of 100.Figure 10.2 shows the percent of students scoring in the top-third, middle third, andlower third of the logical progression scale on individual context-rich examproblems over the two quarters (22 weeks including final exam weeks) of thealgebra-based course. The fluctuations in the graph are due to differences in thelevel of difficulty of the context-rich problems (see Chapter 16 for problemcharacteristics that affect the difficulty of a problem). The general trend, however,is improvement in the logical presentation of students’ problem solutions. A visualinspection of Figure 10.1 shows the improvement in the logical progression of astudent’s solution at the beginning of the course and at the end of the course.Similar improvement trends were found for the other problem-solving skills, withthe exception of General Approach and Appropriate Mathematics. As expected, therewas no improvement in the General Approach scores because each exam tested thenew physics students were learning. We concluded that an instructional approachthat combines the explicit teaching of a problem-solving framework with practiceimplementing the framework in cooperative groups is effective in improving theproblem solving skills of all students on context-rich problems.In addition, two traditional problems were included in the final exams of theexperimental (cooperative problem solving) section and traditional section of thecourse. The results of the problem solving scores for the two sections are shown inFigure 10.3. We concluded that students who are taught an explicit problem-solving

Chapter 10: Results for Partial and Full Implementation of CPS 129Figure 10.3. Comparison of the median students’ scores on two common finalexam problems for CPS and a Traditional section of the algebra-based course.CPS Section:Median(N = 91)TraditionalSection: Median(N = 118)Mann-WhitneyZ StatisticProblem #1 82 62 8.17*Problem #2 71 50 4.20** p

130 Part 2: UsingCooperative Problem Solvingstudents’ conceptual understanding will improve as much as with other innovations,but you may not see a substantial improvement is your students’ problem solvingperformance.Improvement in Knowledge OrganizationOriginal expert-novice research indicated that experts solve unusual physicsproblems by using the cues in the problem to decide what physics principles to apply(e.g., Newton’s second law, conservation of energy). They use this principle(s) toqualitatively analyze the problem situation before beginning a mathematicalsolution. 10 An example of the hierarchical organization of physics knowledge byprinciples is shown in Chapter 2, page //. Novice students, on the other hand, arecued by the surface features of the problem (e.g., free fall, circular motion, inclinedplane) and try to remember the formulas that apply in these cases. They do notqualitatively analyze the problem situation, but immediately start to manipulateequations. An example of the knowledge organization of a novice student is shownin Chapter 2, page //.An early research method was to give experts and novices a set of problems to sortbased on by how they would solve the problem. Our graduate student Ron Keith(now deceased) investigated whether the students who solve context-rich problemsusing the Competent Problem-solving Framework 11 move towards a more hierarchicalorganization of knowledge compared to students in a traditional algebra-basedcourse. 12Interviewees sorted a set of 30 problems, written on cards, into groups based onhow they would solve the problems. Each group was then labeled and described.A small sample of volunteer students from the traditional class, a sample of studentsin the CPS class who were identified as problem-framework users, and a sample ofadvanced physics graduate students and post-docs, participated in this interviewtask. Two examples of the problems are shown in Figure 10.3a.An examination of the labels and descriptions resulted in the identification of twocategories between novices and experts in how they decide to solve a problem, asdescribed below:1. Surface Feature (Novice). No reference to how problem could be solved usingphysics principles (e.g., a "circular motion" label described as "they all havethings going in circles").2. Surface Feature. Refers to application of a specific equation or relationships,but not to physical principles (e.g., a "circular motion" label described as "usecentripetal acceleration, a = v2/r).3. Surface Feature/Physics Principle. Refers to application of a physical law (e.g.,a "collision" label described as "you could solve these problems either byconservation of momentum or energy").Figure 10.4. (a) Two examples of sorting instrument problems. (b) Percentage ofstudents receiving traditional receiving and their weighted categorization scores. (c)

Chapter 10: Results for Partial and Full Implementation of CPS 131Percentage of students who used the CPS problem-solving frameworks and theirweighted categorization scores. (c) Percentage of experts and their weightedcategorization scores.(a) Energy Problem: While on a brief visit to the UMD campus, you park your 1000 kg car atthe top of a 50 ft high, icy hill. It's not your lucky day. Your brakes fail and the car rollsdown the hill. What is the speed of your car when it reaches the bottom of the hill?Kinematics Problem: To reach her destination, a backpacker walked downhill with an averagevelocity of 3 mph. This average velocity resulted because she hiked for 4 miles downhill withan average velocity of 6 mph, then backtracked uphill with an average velocity of 1 mph. Whatdistance did she walk uphill?(b) (c) (d)4. Physics Principle (Expert). Refers to application of physical law (e.g., "forcelaws," "use force diagrams," "energy conservation," "momentumconservation").Most students sorted the 30 problems into a different number of groups based ondifferent categories of how they would solve the problems. We created a weightedaverage score for each interviewee – the category value for each category (1, 2, 3, or4), weighted by the number of problems in the category. Figures 10.3b, 10.3c, and10.3d show the percentage of students with weighted average scores ranging from1.0 to 4.0 in the traditional section, students in the CPS section identified asproblem-framework users, and the experts.A comparison of the graphs in Figures 10.4 b and 10.4c show that the students whouse the problem-solving framework started to move towards expert-likecategorization of problems in just 15 weeks of instruction. This is noteworthyconsidering that it takes several years for students to organize their physicknowledge hierarchically by basic principles. 13

132 Part 2: UsingCooperative Problem SolvingImprovement in Students’ Conceptual Understanding of Motion and Forces asMeasured by the Force Concept Inventory (FCI)The Force Concept Inventory (FCI) is a 30-item multiple-choice test developed byDavid Hestenes, Malcolm Wells, and Greg Swackhamer 14 to probe studentsunderstanding of concepts in mechanics. The test questions are based on interviewswith students by other researchers, the choices reflecting common studentmisconceptions. For decades, physics instructors have used the Force ConceptInventory (FCI) to measure the effectiveness of instructional strategies in improvingstudents’ conceptual understanding of kinematics and Newton’s laws of motion.A standard practice is to administer the multiple-choice test twice, once at the startof the semester and once at the end. Student achievement in conceptualunderstanding is then measured by improvement in the score from pre-instructionto post-instruction. In order to take into account the fact that a 10% improvementfor a student with a 20% correct pre-test score is not necessarily the same as a 10%improvement for a student with a 90% correct pre-test score, a measure called thenormalized gain was developed. The normalized gain in student conceptualunderstanding is defined as follows: g %Correct postinstruction %Correct preinstruction100 %Correct preinstructionThis expression is often referred to in the literature as the ‘‘g’’ or ‘‘Hake’’ factor 15 ,and is the ratio of the actual gain to the maximum possible gain. 16Improvement in FCI Scores with Practice ImplementingCPSFigure 10.5 shows the normalized gain in the Force Concept Inventory (FCI) fordifferent faculty members in the first five years of implementation. The faculty atthe University of Minnesota implemented a partial model of CPS. Cooperativeproblem solving replaced the traditional recitations, and problem-solving labs wereunder development. Each teaching assistant worked with the same students in boththe new discussion sessions and in the labs, and participated in a two-weekworkshop to learn how to implement cooperative problem-solving techniques. Forthe most part, however, the faculty continued with their traditional lectures andapproach to problem solving.Faculty members A, B, C, D, and E taught the course more than one year. In eachcase, their FCI normalized gain scores increased considerably (about 1 – 2 standarddeviations) from the first time they taught the course to the second (or third) timethey taught the course. Faculty Members Z and C (Year 3) are interesting casestudies. Faculty Member Z ignored the initial training of the TAs in CPStechniques. Instead, he met with a student focus group and implemented theirsuggestions in the lectures and discussion sections. The normalized gain in FCIscores was very low ( = 0.10 ± 0.7) – below the national average of 0.20 ± 0.03for traditional lecture courses. Faculty Member C misinterpreted the nature ofgroup problems. Instead of context-rich problems, he gave difficult estimation

Chapter 10: Results for Partial and Full Implementation of CPS 133Figure 10.5. The FCI normalized gain sore for different faculty over the first five-year period ofimplementing partial CPS. The letters under each column represent different faculty members.Figure 10.6. Average pre- and post- instruction Force Concept Inventory scores for the 41faculty members who implemented CPS

134 Part 2: UsingCooperative Problem SolvingFigure 10.7 A comparison of the effect of first-year and mature implementationof an instructional innovation in a calculus-based introductory course on the FCI** The number in parentheses is the number of classes in the samples. The error bars are thestandard deviation of the average of classes for N>1. The institutions in the left labels are: RPI -Rensselaer Polytechnic Institute; 2 Umd - University of Maryland; 17 UMn - University of Minnesota;OSU - Ohio State University; 18 DC - Dickinson College; 6 , 19 and Others -- two other small colleges. 6The source of the data is in the endnotes.problems, which most groups failed to solve. His normalized gain, = 0.27 ±0.5, was slightly higher than the national average for traditional lecture courses.Figure 10.6 shows the pretest and posttest FCI scores for the University ofMinnesota faculty (41 faculty members) for the 17 years we have implemented CPSin the calculus-based course. Incoming student scores on the FCI are slowly rising,probably due to better high school preparation. The partial CPS implementationresults in a post-instruction FCI score of approximately 67%. The best-practiceimplementation of CPS (indicated by arrow on figure) results in a post instructionFCI score of approximately 80%.First Year and Mature Implementation of Different InstructionalInnovationsFigure 10.7 shows the normalized gain on the Force Concept Inventory (FCI) forthe implementation of four research-based instructional innovations in calculusbasedclasses: Interactive Lecture Demonstrations (ILD); Tutorials 20 ; Workshop

Chapter 10: Results for Partial and Full Implementation of CPS 135Physics (WP) 21 ; the partial implementation of Cooperative Problem Solving (CPS);and the best-practice implementation of CPS. For the other institutions in theFigure 10.4, the partial CPS implementations and Tutorials consist of replacing astandard recitation session with a CPS session or Tutorial session, while the lecturesand labs remain the same. The best-practice CPS for calculus-based classes wasimplemented by one of the developers (author KH).Figure 10.7 shows that the first-year implementation of Interactive LectureDemonstrations (ILD), Tutorials, and the partial CPS implementations ( = 0.35± 0.3) improves students’ conceptual understanding of forces by 15 % compared totraditional instruction ( = 0.20 ± 0.03). Mature implementation of partial CPS( = 0.44 ± 0.03) improves students’ conceptual understanding an additional 9%.Implementation of best-practice CPS ( = 0.59 ± 0.04) further improves FCIgains by an additional 15%. 22 This may be approaching the limit of what can beexpected at a university that enrolls 150-200 students in each section with 5-7teaching assistants.Workshop Physics (WP) yielded the largest improvement in students understandingof forces for both first-year implementation ( = 0.41 ± 0.02) and for matureimplementation ( = 0.74). The ability to integrate lecture, labs, and recitationinto one classroom with an excellent curriculum and pedagogy accounts for thisdifference for colleges with small class sizes.Improvement in Students’ Conceptual Understanding of Motion and Forces asMeasured by Open-Response Written QuestionsThe Ramp ProblemThe open-response Ramp Problem (Figure10.8a) is a written adaptation of aninterview question developed by Lillian McDermott and the Physic EducationGroup at the University of Washington. 23 The problem is designed to probestudents’ understanding of acceleration. It was administered pre- and postinstruction to an algebra-based course the year before we implemented best-practiceCPS. At the same time, it was administered to a traditional section of the calculusbasedcourse. Our graduate student, Jennifer Blue, analyzed the results. 24Description of Categories. Standard qualitative research techniques were used tocategorize the written responses. 25 Three major categories of responses emergedfrom the analysis: responses that include the accepted ideas about acceleration,responses that include alternative ideas (two sub-categories), and responses thatcould not be coded because too little was written. Figure 10.7b shows thepercentage of students in each category, pre- and post-instruction, in the algebrabasedpartial CPS class and in the traditional calculus-based class. (Uncertainties inthe results are estimates of the sampling error calculated as (pq/N)1/2. The resultsare reported for students who responded to the problem both pre-instruction andpost-instruction.

136 Part 2: UsingCooperative Problem SolvingFigure 10.8a. The Ramp Question1Motion Up the Ramp23Motion Down the Ramp 2A steel ball is launched with some initial velocity, slows down as it travels upa gentle incline, reverses direction, and then speeds up as it returns to itsstarting point. Assume friction is negligible.a. Suppose we calculated the acceleration of the ball as it's moving up the ramp(from 1 to 2), and the acceleration as it's moving down the ramp (from 2 to3). How would these two accelerations compare? (i.e., Are the accelerationsthe same size? The same direction?) Explain your reasoning.b. Does the ball have an acceleration at its highest point on the incline (atposition 2)? Explain your reasoning.Figure 10.8b. Types of responses to question: How does the accelerationcompare up and down the ramp?Category of ResponseAlgebra-basedPartial CPS(N = 112)Pre(%)Post(%)Calculus-basedTraditional(N = 138)Pre(%)Post(%)1. Includes accepted idea of6 ± 3 79 ± 4 19 ± 4 40 ± 5acceleration2. Includes alternative ideas:a. confuse v and a, butbelieve motion up and down 58 ± 5 16 ± 4 57 ± 5 51 ± 5are the sameb. confuse v and a, but believemotion up and down are not 35 ± 5 2 ± 1 17 ± 4 6 ± 3the same3. Response cannot be coded 1± 1 3 ± 2 7 ± 3 3 ± 2Results. Figure 10.8b shows that twice as many students in the partial CPS algebrabasedclass (79%) had an understanding of acceleration at the end of the course thanthe students in the traditional calculus-based class (40%). 57 percent of the calculusbasedstudents still confused velocity and acceleration at the end of the first quarter(10 weeks), compared to only 18% of the algebra-based students. The results forthe traditional calculus-based students are consistent with the interview results byTrowbridge and McDermott 17 -- out of a sample of 36 interviewed students, 64%were successful after instruction. The results for the full CPS model are alsoconsistent with the Force Concept Inventory (FCI) results in the last section.

Chapter 10: Results for Partial and Full Implementation of CPS 137The Accelerating Car QuestionsWe designed the Accelerating Car Questions, shown if Figure 10.9a, to probestudents’ understanding of the nature of forces and Newton’s second law. Thequestions were administered to students in an algebra-based course the year beforewe implemented best-practice CPS (first data column in Figures 10.9b and 10.9c).Later, the same questions were administered to students in a traditional section ofthe calculus-based course (second data column). Later still, the questions wereadministered at the beginning of a second semester, when about half of the studentscame from other partial CPS models sections, and half from the full CPS model(third data columns). Jennifer Blue 18 completed the analysis for the algebra basedcourse and Tom Foster completed the analysis for the calculus-based course. 26Description of Categories for Nature of Forces. Again, the students’ responseswere examined using standard qualitative research techniques. 19 Four categoriesemerged for the nature of forces. In the first category only Newtonian forces wereidentified and labeled, although other parts of the responses could be wrong. Thesecond response category included Newtonian forces, but at least one was a 3rd-lawpair force on the wrong object. For example, a student drew “the force of the carseat on the passenger” on their free-body diagram of the car. The third categoryincluded incorrect “pseudoforces,” such as “the force of acceleration of the car” and“force of the car’s engine that turns the tires.” The fifth category contains studentresponses with no forces drawn.Results for the Nature of Forces. Figure 10.9b shows that on the pretest, the samepercentage of students in the algebra-based partial CPS course drew Newtonianforces (12%) as in the calculus-based traditional course (14%). The algebra-basedpartial CPS students made some progress in their understanding of the nature offorces; on the post-test 63% of the students drew Newtonian forces, compared with12% on the pretest. 54 percent of the students in the traditional calculus-basedcourse drew Newtonian forces on the post-test.For students in a calculus-based course, participation in a partial CPSimplementation results in 8% more students drawing Newtonian forces at the end ofthe course (62%) compared to the traditional course (54%). Participation if bestpracticeCPS results in an additional 13% of students drawing Newtonian forces(75%). This may be near the limit of what can be expected in large enrollmentcourses.Description of Categories for Newton’s Second Law. The analysis of studentunderstanding of Newton’s Second Law (Figure 10.9c) was separate from theanalysis of student understanding of forces, so some student responses in highercategories may contain “pseudoforces.” The first category contains responses thatinclude accepted ideas about the sum of forces, although other parts of theresponses could be wrong.

138 Part 2: UsingCooperative Problem SolvingFigure 10.9a. The problem about an accelerating carYou are a passenger in a car that is traveling on a straight road whileincreasing speed from 30 mph to 55 mph. You wonder what forces cause you andthe car to accelerate. When you pull over to eat, you decide to figure it out.…a. On the picture, draw and label arrows (vectors)representing all the forces acting on the car while itis accelerating. The length of the arrows shouldindicate the relative sizes of the forces (i.e., a larger force should berepresented by a clearly longer arrow, equal forces by arrows of equallength). Below the picture, describe in words each force shown.b. Which force(s) cause the car to accelerate? Explain your reasoning.Figure 10.9b. Types of responses for the nature of the forces on the carCategory of ResponseAlgebra CoursePartial CPS(N = 112)pre(%)post(%)Baseline:Traditional(N = 100)pre(%)Calculus Coursepost(%)PartialCPS(N=85)post(%)Best-PracticeCPS(N=71)post(%)1. Only Newtonian forces 10 ± 3 51 ± 5 10 ± 3 39 ± 5 58 ± 5 73 ± 42. Newtonian forces, but some are3 rd -Law pair forces drawn on wrong 2 ± 1 12 ± 3 4 ± 2 15 ± 4 4 ± 2 2 ± 1object3. Include non-Newtonian or“ pseudoforces ” (e.g., force ofacceleration; force of the78 ± 4 37 ± 5 73 ± 4 39 ± 5 38 ± 4 25 ± 4engine)4. Cannot be coded 10 ± 3 0 8 ± 3 1 ± 1 0 0Figure 10.9c. Types of responses to the question: Which force(s) cause the car to accelerate?Algebra CourseCalculus CourseCategory of ResponsePartial CPS(N = 112)pre(%)post(%)BaselineTraditional(N = 100)pre(%)post(%)Partial.CPS(N=85)post(%)Best-PracticeCPS(N=71)post(%)1. Includes correct ideas aboutsumming Newtonian forces2. Vague or incorrect summing3. Includes Alternative Ideasa. Attributes acceleration toonly one forceb. Attributes acceleration tosomething other than a forceon the diagram4. Can not be coded3 ± 2 27 ± 5 7 ± 3 20 ± 4 42 ± 5 58 ± 424 ± 5 9 ± 325 ±429 ± 5 24 ± 4 21 ± 313 ± 4 56 ± 5 4 ± 2 23 ± 4 21 ± 4 19 ± 332 ± 5 048 ±527 ± 5 8 ± 3 16 ±423 ± 4 7 ± 3 07 ± 3 6 ± 2 4 ± 2

Chapter 10: Results for Partial and Full Implementation of CPS 139The second category includes responses that contain vague or incorrect ideas abouta “sum of something.” Some responses in this category list forces without saying,“sum.” Other responses in that category list things that are not forces, nor are theyon the free-body force diagrams, like coefficients of friction. The third categoryincludes responses that do not sum; they incorrectly cite only one force as the causeof acceleration. Most students in this category had drawn unbalanced forces ontheir drawings. The fourth category includes responses that cite things other thanforces on the diagrams as the cause of acceleration. Some responses in this categoryattribute the car’s acceleration to its engine, and others cite things like “motion” and“momentum” as the cause of acceleration. Responses in the fifth category could notbe coded.Results for the Newton’s Second Law. Figure 10.9c shows that students in both thealgebra-based partial CPS course and the traditional course showed very littleimprovement in their understanding of Newton’s second law from pretest toposttest (23% and 13% respectively). For students in a calculus based course,participation in a the partial CPS course results in 22% more students with thecorrect idea of summing forces at the end of the course (42%) compared to thetraditional course (20%). Participation if the best-practice CPS course results in anadditional 16% of students with the correct idea of summing forces (58%). Theseresults indicate that there is room for further improvement.Summary for FCI and Open-Ended QuestionsThe results for the improvement in students’ conceptual understanding of motionand forces as measured by the Force Concept Inventory (FCI) are consistent withthe results for the open-response written questions. For the calculus-based course,partial implementation of CPS improved students conceptual understanding ofmotion and forces for the FCI ( = 0.44 ± 0.03 versus = 0.20 ± 0.03 fortraditional courses), students’ understanding of the nature of forces (gain of 8% ofstudents drawing Newtonian forces compared to traditional), and students’understanding of Newton’s second law (gain of 22% of students with correct idea ofsumming forces compared to traditional). Implementation of best-practice CPSresulted in the most improvement on the FCI ( = 0.59 ± 0.04 versus =0.20 ± 0.03 for traditional courses), conceptual understanding of the nature offorces (gain of 21%), and understanding of Newton’s second law (gain of 38%).Improvement in Students’ Learning AttitudesStudents enter our courses with sets of attitudes and beliefs about learning science.How we conduct our class sends messages about how, why, and by whom science islearned. Such messages are being studied with the goal of developing more expertlikeviews on the nature and practice of science in our students. 27 [

140 Part 2: UsingCooperative Problem SolvingFigure 10.10. Learning attitudes for a best-practice implementation of CPSOver the last two decades, physics education researchershave developed several survey instruments to measure theseattitudes and beliefs and to distinguish the beliefs of expertsfrom the beliefs of novices. 28 For example, expertphysicists see physics as a coherent framework of conceptsthat describe nature and are established by experiment.Novices see physics as isolated pieces of information withno connection to the real world, which are handed down byauthority (e.g. teacher) and must be memorized.Data have shown that, traditionally, student beliefs become more novice-like overthe course of a semester. 29 Even in courses using reformed classroom practices thatare successful at improving student conceptual learning of physics, student beliefstend not to improve. Some success has been achieved, however, in coursesspecifically designed to attend to student attitudes and beliefs. 30At the University of Minnesota, the Colorado Learning Attitudes about ScienceSurvey (CLASS Version 3) was used in 2009 to measure student beliefs at the start(pre) and end (post) of an introductory physics courses for pre-medicine and biologymajors. This course implemented the best-practice CPS. The ‘All Categories’favorable score is measured as the average percentage of the 42 survey statements towhich the students answer in the favorable sense (e.g. as an expert physicist would).The survey is also used to measure specific belief categories by looking at subsets ofstatements. Included are measurements of the following categories: ‘PersonalInterest’ (I think about physics in my life); ‘Real World Connections’ (physicsdescribes the world); “Problem Solving General” (equations represent concepts);Problem Solving Confidence’ (I can usually figure out a way to solve problems);Problem Solving Sophistication (When I get stuck, I can figure out a differentmethod); ‘Sense Making/Effort’ (I put in the effort to make sense of physics ideas);‘Conceptual Understanding’ (physics is based on a conceptual framework); and“Applied Conceptual Understanding’ (After I study a topic, I can apply it).

Chapter 10: Results for Partial and Full Implementation of CPS 141The results, shown in Figure 10.9, indicate significant improvement in all categoriesof students’ attitude about learning physics. The overall improvement was 11% offavorable responses. This improvement is larger than in courses reported to attendto student attitudes and beliefs. We concluded that establishing a culture of expertpractice (see Chapter 6), demonstrating a research-based problem-solvingframework when solving problems in lectures, requiring students to practice usingthe framework for solving context-rich problems in cooperative groups (and forhomework), appropriate course grading, and providing appropriate scaffolding,improves students’ attitudes towards learning physics.Endnotes1 Heller, P., Keith, R., & Anderson, S. (1992). Teaching problem solving throughcooperative grouping. Part 1: Groups versus individual problem solving, AmericanJournal of Physics, 60(7), 627-636.2 For example, if students have a persistent misconception about a principle orconcept (e.g., Newton’s second law or acceleration), they would not typically beable to solve a complex problem correctly. On the other hand, students may beable to answer conceptual or qualitative questions about a principle or concept,but not be able to apply the concept correctly to a quantitative problem situation.They may not know the conditions of applicability of the principle or concept.Or they may have disconnected qualitative and quantitative knowledge, and applythe plug-and-chug or pattern-matching strategy to all quantitative problems,without regard to the meaning of principles and concepts.3 Cummings, K., Marx, J., Thorton, R., & Huhl, D. (1999). Evaluating innovationin studio physics, American Journal of Physics, Physics Education Research Suppl.,67(7), S38-S44.4 The defining characteristics of Studio Physics are integrated lecture/laboratorysessions, small classes of 30–45 students, extensive use of computers,collaborative group work, and a high level of faculty–student interaction. Eachsection of the course is led by a professor or experienced instructor, with helpfrom one or two teaching assistants. The teaching assistants’ roles are to circulatethroughout the classroom while the students are engaged in group work.5 Interactive Lecture Demonstrations (ILD) was developed by David Sokoloff andRon Thornton to guide students through a three-part learning sequence ofpredict, confront, and resolve. The instructor shows and explains a situationdesigned to conflict with their predictions, gathers the students’ predictions, anddemonstrates and takes data as students copy the result. The students are thenasked for an explanation (confront), and the instructor makes the bridge tosimilar situations (resolve). See Sokoloff, D.R., and Thorton, R.K. (1997), Usinginteractive lecture demonstration to create an active learning environment, PhysicsTeacher, 35, 340-347; and Sokoloff, D.R., and Thorton, R.K. (2001), InteractiveLecture demonstrations, NY: Wiley

142 Part 2: UsingCooperative Problem Solving6 See Dufresne, R.J., Gerace, W.J., Hardiman, P.T., & Mestre, J.P. (1992),Constraining novices to perform expert-like problem analyses: Effects on schemaacquisition, Journal of Learning Science, 2, 307-321; and Leonard, W.J., Dufresne,R.J., and Mestre, J.P. (1996), Using qualitative problem-solving strategies tohighlight the role of conceptual knowledge in solving problems, American Journalof Physics, 64, 1495.7 See Antonenko, P.D., Jackman, J., Kumsaikaew, P., Marathe, R.R., Niederhauser,D.S., Ogilvie, C. A., and Ryan, S.M., (submitted for publication), Understandingstudent pathways in context-rich problems, retrieved November 24, 2010,http://www.public.iastate.edu/~cogilvie/edresearch.html.8 For a brief discussion of these curricular innovations, see Hsu, L., Brewe, E.,Foster, T. M., & Harper, K. A. (2004), Resource letter RPS-1: Research inproblem solving. American Journal of Physics, 72(9), Section IV9 See Taconis, R., Ferguson-Hessler, M.G.M., &. Broekkamp, H. (2001), Teachingscience problem solving, Journal of Research in Science Teaching, 38, 442–468. Theauthors did a meta-analysis of 22 previously published articles on teachingproblem solving in science classes.10 See Chi, M. T. H., Feltovich, P. J., & Glaser, R. (1981), Categorizations andrepresentation of physics problems by experts and novices, Cognitive Science, 5,121-152; Larkin, J., McDermott, J., Simon, D., & Simon, H. A. (1980), Expertand novice performance in solving physics problems, Science, 208, 1335-1342; andHardiman, P.T., Dufresne, R., and Mestre, J.P. (1989), The relation betweenproblem categorization and problem solving among novices and experts, Memoryand Cognition, 17, 627.11 We developed criteria for selecting problem-framework users and non-users byexamining the solutions of all the students for three context-rich problems onefrom the third exam, and two from the final (after 10 weeks of the quarter). Thecriteria for including students in user group did not include solving problemscorrectly, but on the qualitative analysis of the problem and planning a solution.12 Keith, R. (1993), Correlation between the consistent use of a general problemsolvingstrategy and the organization of physics knowledge. Unpublished PhDthesis: University of Minnesota.13 The opposite has also been shown. That is, novices who classify problems in amore expert-like manner are more proficient in solving problems. See Hardiman,P.T., Dufresne, R., &Mestre, J.P. (1989). The relationship between problemcategorization and problem solving among experts and novices, Memory andCognition, 17, 627–638.14 Hestenes, D., Wells, M., & Swackhammer, G. (1992). Force concept inventory,The Physics Teacher, 30(3), 141-158

Chapter 10: Results for Partial and Full Implementation of CPS 14315 Hake, R. (1998). Interactive-engagement vs traditional methods: A six thousandstudent survey of mechanics test data for introductory physics courses. AmericanJournal of Physics, 66: 64-7416 There is a question about whether relative gain is the best measure ofimprovement, especially in a class that has a high pretest average. A commonpractice today is to leave out the top 10-15 percent of the students on the pretest,and report the absolute gain for those students in the class who can improvetheir FCI scores significantly.17 Redish, E.F., Saul, J.M., & Steinberg, R.N. (1997). On the effectiveness, of activeengagement microcomputer-based laboratories, American Journal of Physics, 65 (1),45-5418 Communication from Jeff Saul, 1996.19 The for Dickinson College, mature implementation is from Redish, E.F.(2003). Teaching physics with the physics suite, MA: Wiley20 Tutorials were developed by Lillian McDermott and the Physics Education Groupat the University of Washington. This supplementary curriculum is designed tobe used in small group sessions in which three or four students work togethercollaboratively. Worksheets guide students through a three-part learningsequence of predict, confront, and resolve. See McDermott, L.C. and Shaffer,P.S., (2002). Tutorials in introductory physics and homework package, Upper SadlerRiver NJ; Prentice Hall21 Workshop Physics instruction is based on a four-part learning sequence. Studentsin small classes of 10 – 20 students make predictions about a phenomenon,reflect on their observations and try to reconcile any differences; they developdefinitions and equations from theoretical considerations; they performexperiments to verify predictions based on theory; they apply their understandingin solving problems. See Laws, P.W. (1991), Calculus-based physics withoutlectures, Physics Today, 44(12), 24-31; Laws, P.W. (1997). Workshop physics activityguide, NY, Wiley; and Laws, P. (1997), Millikan lecture 1996: Promoting activelearning based on physics education research in introductory physics courses,American Journal of Physics, 65, 14-21.22 The full CPS model was only implemented once in the calculus-based coursebecause different sections of the course (~ 1000 students) are taught eachsemester. Students can enroll in any section, and take a common final exam.Therefore, the sections cannot be very different from each other in content orpedagogy.23 Trowbridge, D.E. & McDermott, L. C. (1981). Investigation of studentunderstanding of the concept of acceleration in one dimension, American Journalof Physics, 49(3), 242-253.

144 Part 2: UsingCooperative Problem Solving24 Blue, J. (1993). Sex differences in open-ended written response questions aboutmotion and forces, Unpublished Master’s Degree Paper, University of Minnesota.25 See Strauss, A.L. (1987). Qualitative analysis for social scientists, Cambridge:Cambridge University Press; and Strauss, A.L. and Corbin, J. (1990). Basics ofqualitative research: grounded theory procedures and techniques, Newbury Park, CA: Sage.26 Foster, T. & Heller, P. (1996). Does continuing in an introductory Physicssequence help students understand the basics? A peek into Pandora’s Box, AAPTAnnouncer, 26(2): 6127 See Hammer, D. (2000), Student resources for learning introductory physics.American Journal of Physics, Physics Education Research Supplement, 68(S1), S52-S59; Elby, A. (2001), Helping physics students learn how to learn, AmericanJournal of Physics, Physics Education Research Supplement 69, S54-S64 (2001); andRedish, E.F, (2003), Teaching Physics with Physics Suite, Wiley.28 See Redish, E.F., Saul, J.M., & Steinberg, R.N. (1998), Student expectations inintroductory physics, American Journal of Physics, 66, 212-224; Halloun, I. (1996).Views about science and physics achievement, The VASS story, in Proceedings of theInternational Conference on Undergraduate Physics Education, American Institute ofPhysics Press, College Park, MD; Elby, A. (2001), Helping physics students learnhow to learn, American Journal of Physics, Physics Education Research Supplement 69,S54-S64; and Adams, W.K., Perkins, K.K., Dubson, M., Finkelstein, N.D., andWieman, C.E. (2005), The design and validation of the Colorado LearningAttitudes about Science Survey, Proceedings of the 2004 Physics Education ResearchConference, Rochester, NY: PERC Publishing.29 Redish, E.F., Saul, J.M., & Steinberg, R.N. (1998). Student expectations inintroductory physics, American Journal of Physics, 66, 212-22430 See Redish, E.F. (2003), Teaching Physics with Physics Suite, Wiley; Hammer, D.(2000), Student resources for learning introductory physics. American Journal ofPhysics, Physics Education Research Supplement, 68(S1), S52-S59; and Perkins, K.Adams, W. K., Finkelstein, N. D., Pollock, J. S., & Wieman, C. E. (2005),Correlating student attitudes with student learning using the Colorado LearningAttitudes about Science Survey, Proceedings of the 2004 Physics Education ResearchConference, Rochester, NY: PERC Publishing.

Part145

146In this part . . .Another cliché is that “the devil is in the details.” We are assuming you havemade the decision to teach a CPS session with your students. The chapters inthis part provide detailed descriptions or how to prepare for a CPS session,teach the session, and monitor, diagnose, and intervene with groups.Chapter 11 provides a description of how to prepare for a CPS session,including assigning students to groups with roles or rotating roles, preparing agroup problem and information sheet, writing the problem solution, preparingan answer sheet (optional), and preparing a group function evaluation sheet (asnecessary).Chapter 12 explains the steps in teaching a CPS session, including openingmoves, the middle game, and the end game.Chapter 13 provides information about how to monitor, diagnose, and intervenein groups, particularly dysfunctional groups and groups having difficulty withphysics.

147Chapter 11Preparation for CPS SessionsIn this chapter: Overview of the routine for a CPS session. How to assign your students to groups. Preparing a group Problem & Information sheet and deciding on the part of the solution youwant groups to discuss. Preparing an Answer Sheet (optional). Preparing a Group Functioning Evaluation form (optional)The usual Cooperative ProblemSolving (CPS) routine, like a gameof chess, has three parts -- OpeningMoves, a Middle Game, and an End Game.As in chess, both the opening moves andthe end game can be planned in detail.The middle game - collaborative problemsolving -- has many possible variations.Opening Moves (~ 5 minutes). Opening moves determine the mind set thatstudents should have during the Middle Game -- the collaborative solving of aproblem. The purpose of the opening moves is to answer the following questionsfor students. Why has this particular problem been chosen? What should we be practicing and learning while solving this problem? How much time will we have? What is the product we should have at the end of this time?Educational research indicates that providing students this simple informationbefore they start leads to better learning and higher achievement. An example of anopening move is shown in Figure 11.1.Middle Game (~ 35 minutes). This is the learning activity -- students workcollaboratively to solve the problem. During this time, your role is one of coach.You circulate around the room, listening to what students in each group are sayingand observing what the Checker/ Recorder is writing. You intervene when a groupneeds to be coached on an aspect of physics or is not functioning well. At the endof the allotted time, you have your groups draw and write on the board the parts ofthe solution that you specified in your opening moves.

148 Part 3: Teaching a CPS SessionExample of Opening MovesFigure 11.1. Examples for beginning and ending a CPS session.We have been studying the conservation principles in class -- the conservation of energyand the conservation of momentum. The problem you will solve today was selected to helpyou learn when and how to apply these principles.You will have 35 minutes to work on the problem. At the end of that time, you will beasked to draw your diagrams and list the equations you used to solve the problem on theboard.Example End-game QuestionsLook at the momentum vector diagrams on the board. How are they the same and how arethey different?Is there different physics represented in the diagrams, or the same physics?Look at the diagrams for group #1 and #5. What is missing in these diagrams?Does the order -- x direction first or y direction first -- make any difference to the finalsolution?End Game (~ 10 minutes). The end game determines the mind-set students havewhen they leave the class -- do they think they learned something or do they think itwas it a waste of their time. The purpose of the end game is to help students answerthe following questions. What have I learned that I didn't know before? What did other students learn? What should I concentrate on learning next?A good end game helps students consolidate their ideas and produces discrepanciesthat stimulate further thinking and learning. Typically the instructor gives students afew minutes to examine what each group produced, then leads a whole-classdiscussion of the results. Your role as the instructor is to facilitate the discussion,making sure students are actively engaged in consolidating their ideas.There are several decisions to make and materials to prepare before teaching a CPSsession, as shown outlined below.1. Assign students to groups (if changing groups) and assign/rotate a role for eachstudent. Make a copy of the group and role assignments (for preparing anoverhead or for copying on the board before class begins).2. Write/Adapt a context-rich problem that meets the criteria for a good groupproblem (see Chapter 15).

Chapter 11: Preparation for a CPS Session 1493. Write a problem solution following the problem-solving framework that youhave decided to teach your students. Decide what part of the solution you wantyour groups to write on the board (or butcher paper, whiteboards) during class.4. Make a computer projection or photocopy of the problem and informationsheet for examples see pages // and //), and photocopies of a group AnswerSheet (optional) that contains the major cues for the problem-solving frameworkyou are teaching (example on pages //-//).Each of these preparations is discussed in the sections of this chapter.Assign Students to Groups with RolesHow many groups will I have?The optimal group size is three. For example, suppose you had 17 students in adiscussion class. Then you would have 5 three-member groups and two studentsleft over.We have found that groups of four usually work better than pairs (see Chapter 8).So with seventeen students, you would normally assign students to five groups, threegroups with three members and two four-member groups.What criteria do I use to assign students to groups?There are three criteria we use to assign students to groups.1. Problem-solving Performance. The most important criterion for assigningstudents to groups is their problem solving performance based on past problemsolvingtests. That is, a three-member group would ideally consist of a higherperformance,a medium-performance, and a lower-performance student. Fourmembergroups would ideally consist of a high performance, medium-highperformance, medium-low performance, and a low-performance student. [SeeChapter 9 for some research support forthis criterion.] There are two other"rules of thumb" for assigning studentsto groups. These rules should bemodified by your knowledge of thesocial interactions of your students.2. Gender. Our observationsindicated that frequently groups withonly one woman do not function well,especially at the beginning of class. Tobe on the safe side, avoid groups withGender We observed a group,consisting of a lower-performanceman, a medium performance man anda high performance woman, having avigorous discussion about the path ofa projectile. The men insisted on apath following the hypotenuse of atriangle; while the woman argued forthe correct parabolic trajectory.At one point, she threw a penhorizontally, commenting as it fell tothe floor, "There see how it goes. Itdoes not go in a straight line!" Evenso, she could not convince the twomen, who politely ignored her.

150 Part 3: Teaching a CPS Sessiononly one woman. We found the difficulty is with the men, not the women (seeexample on previous page). Regardless of the strengths of the lone woman, manymen in our groups tend to ignore her. On the other hand, it is not a good idea toassign all the students in a class to same-gender groups. The women notice and tendto suspect gender discrimination. Curiously, no one seems to notice when all mixedgendergroups have two women.3. English as a Second Language (ESL). Students from other cultures oftenhave difficulty adjusting to group work, especially in mixed-gender groups. Theirdifficulties are exacerbated if English is their second language (ESL). So to be onthe safe side, whenever possible we assign ESL students to same-gender groups ofthree.Prepare Problem & Information SheetThe success of CPS depends on designing group practice problems that place“fences” or barriers on all paths that do not involve using a logical and organizedproblem-solving strategy. As described previously in Chapter 3, context-richproblems are specifically designed so that: It is difficult to manipulate a formula to get an answer. It is difficult to match an example solution pattern to get an answer. It is difficult to solve the problem without first analyzing the problemsituation. Solution patterns are not cued by physics words such as “inclined plane,”“starting from rest,” or “inelastic collision”. Logical analysis using fundamental physics concepts is reinforced.In addition, to be effective a context-rich problem should have an appropriate level ofdifficulty for its intended use, in this case as a group practice problem (see Chapter16).Below are some suggestions for preparing an appropriate problem and informationsheet.Step . The first step is to decide on yourgoal(s) for the group practice problem. For example,suppose you have finished studying elastic collisionsand introduced inelastic collisions in your lecture.Your students, however, have not had time tocomplete the more difficult homework problems oninelastic collisions. In this case the group practiceproblem should involve an inelastic collision, butshould not be too difficult (i.e., involve vectors or both inelastic collisions and thetransfer of momentum). You also know that your students are still having difficulty

Chapter 11: Preparation for a CPS Session 151Figure 11.2. The context-rich Skateboard Problem.You are helping your friend prepare a skateboard exhibition. The idea is for your friend totake a running start and then jump onto a heavy duty 15-lb stationary skateboard. Your friend,on the skateboard, will glide in a straight line along a short, level section of track, then up asloped concrete wall. The goal is to reach a height of at least 6 feet above the starting pointbefore rolling back down the slope. The fastest your friend can run and safely jump on theskateboard is 20 feet/second. Can this program work as planned? Your friend weighs in at125 lbs.distinguishing between conservation of momentum and conservation of energy. Toconfront this difficulty, you want a problem that requires both the conservation ofmomentum and the conservation of energy for a solution.Step . The next step is to adapt a context-rich problem to meet your goal forthe problem. Many textbooks have context-rich problems that can be adapted forgroup work. This book contains some context-rich problems (See Appendices Band C). Some problems are also available on our website(http://groups.physics.umn.edu/physed/Research/CRP/crintro.html). Theskateboard problem, shown in Figure 11.2, is from Appendix B. If you cannotfind and adapt a context-rich problem, you can adapt a traditional textbookproblem, following a procedure described in Chapter 16.Solve your problem and check that it is the right level of difficulty for a practicegroup problem (see also Chapter 14), using the Criteria shown in Figure 11.3. Asthis figure shows, the skateboard problem is suitable for a practice group problem,since it meets all the criteria.If you know how to arrive at a solution, even if you don’t know the answer, then thequestion is not a problem for you (see Chapter 4). Because you are an expert, the“problems” in an introductory physics course are not real problems for you, soyou don’t use a problem-solving framework to solve them. But they are problemsfor your students.Step . The final step is to prepare an information sheet. The informationsheet contains all constants and "equations" they need to solve the problem, asillustrated in Figure 11.4 (see also Chapter 3, pages //-//). The use of aninformation sheet is another example of applying the 3rd Law of Instruction --Make it easier for students to do what you want them to do and more difficult to do what youdon't want. In this case, we don't want students to spend time searching a textbook(or their memory) for appropriate equations or a matching example problem, sowe supply them with the equations. Then groups spend their time discussing whatthe equations mean and how the concepts and principles should be applied tosolve the problem. The information sheet "grows" with the course -- nothing istaken off, but new constants, fundamental principles, and concepts are added asmore is learned.

152 Part 3: Teaching a CPS SessionFigure 11.3. Criteria for a good group problem.Criteria for Group ProblemSkateboard Problem A group problem must be designed so that: There is something to discuss initially so thateveryone (even the weakest member) cancontribute to the discussion. There are several decisions to make insolving the problem.Students need to spend time initially drawinga picture of the situation.Students must decide what assumptions tomake and what their target variable will be. A group problem must be challenging enough so that: Even the best student in the group can notimmediately see how to solve the problem,and all students feel good about their role inarriving at a solution. Knowledge of basic physics concepts isnecessary to interpret the problem. Students' alternative conceptions about thephysics naturally arise and must bediscussed.The problem cannot be solved by substitutionof known values into momentum or energyequations.Students must apply the conservation ofmomentum and the conservation of energy.Students must understand the differencebetween conservation of energy andmomentum, and when it is appropriate to usethese principles. At the same time, the problem must be simple enough so that: The mathematics is not excessive orcomplex. The solution path, once arrived at, can beunderstood, appreciated, and easilyexplained to all members of the group. A majority of groups can reach a solution inthe time allotted.The problem requires only simple algebra.Once students have decided when and how toapply the conservation of momentum andenergy, the solution is straightforward.Figuring out how to solve the problem, whichtakes the most time, can be done in the timeallotted (about 35 minutes).Write the Problem SolutionIn most CPS sessions (except for a group exam problem), students may not havetime to complete the problem solution before you stop them to conduct thewhole-class discussion. This makes some students anxious or uncomfortable, andit is very difficult to get groups to stop solving the problem. Because the classdiscussion cannot go over the entire problem solution, many students needreassurance that their group solution is correct (or at least on the right track).Anxiety is relieved when they know they will see a complete solution after thesession (usually posted on you class web page). This makes it easier to stop thegroups and have them participate in the whole-class discussion.

Chapter 11: Preparation for a CPS Session 153Figure 11.4. Example list of Useful Information: Calculus-based courseUseful Mathematical Relationships:For a right triangle: sin = a c , cos = b c , tan = a b ,a 2 + b 2 = c 2 , sin 2 + cos 2 = 1For a circle: C = 2R, A = 2For a sphere: A = 4R 2 , V = 4 3 R3If Ax 2 + Bx + C = 0, then x = B B2 4AC2A; d(zn )dz nzn1Fundamental Concepts and Principles:v x av xts ave distta x av v xtE f E i E in E outE transferp f p i p in p outp transferv x dxdts drdta x dv xdtKE 1 2 mv2 p mvddt v rF x ma x F 12 = F 21 E transfer F x dxp transfer F dtUnder Certain Conditions:x f 1 2 a x t 2 v ox t x o ,a r v2r , F k F N , F s F N , F kx, PE mgy,PE 1 2 kx2 ,p transfer F parallel tUseful constants: 1 mile = 5280 ft = 8/5 km, g = 9.8 m/s 2 = 32 ft/s 2Step . The first step is to write the problem solution using the specificproblem-solving framework that you have decided to teach your students (seeChapter 4 and Chapter 14). An example solution for the Skateboard Problem,following the Competent Problem-solving Framework: Calculus Version, is shown inFigure 11.5. In our experience writing problem solutions is difficult forinstructors because the solution shows how you would solve the problem if youdid not already know how to solve it. This is a difficult mindset for expertproblem solvers to assume.

154 Part 3: Teaching a CPS SessionThe specific framework you decide to teach your students in an introductorycourse should be a logical and organized guide through the solution of a problem.It gets them started, guides them to what to consider next, organizes theirmathematics, and helps them determine if their answer is correct. It does not alwaysyield the most elegant solution but it is straightforward, easy to understand, and very general.Step . The next step is to decide what part of the problem solution you wanteach group to write on the board for the whole-class discussion. Typically, youselect part(s) of the solution that you know are difficult for your students. Forexample, you know that students have difficulty distinguishing the conservation ofenergy from the conservation of momentum. So for the skateboard problem, youcould have groups write on the board their physics diagrams/representations ofthe problem, in this case the momentum and energy descriptions of the problem.Prepare An Answer Sheet (Optional)If you are following the grading recommendations from Chapter 9, you may wantto provide an Answer Sheet that contains specific cues for the problem-solvingframework you are teaching your students. An example answer sheet for thecalculus-based Competent Problem-solving Framework is shown in Chapter 5, page //.Notice that the skateboard problem example solution is written on this answersheet.Prepare Group Functioning Evaluation Sheets (as needed)One of the elements that distinguish cooperative groups from traditional groups isthe opportunity for students to discuss how well they are solving the problemstogether and how well they are maintaining effective working relationships amongmembers. We found that our students needed to evaluate their group functioningfairly often at the beginning of a course. For this purpose, we used the GroupFunctioning Evaluation form shown in Chapter 9, page //. After students arecomfortable working in groups and know how to co-construct group solutions, theytypically need only an occasional opportunities to evaluate their group functioning.After the first 5 - 6 weeks of the introductory course, we use two rules of thumb todecide if students need to discuss their group functioning.1. Change to New Groups. Typically groups are more effective when theyevaluate their functioning the first time they work together as a new group.

Chapter 11: Preparation for a CPS Session 155Figure 11.5. Example Solution of Skateboard Problem Following theCompetent Problem-solving Framework for Calculus-based CourseFOCUS on the PROBLEMPicture and Given Informationvo = 20 ft/sec v 1 = ? v 2 = 0h = 6 ftWr = 125 lbs (weight of runner, R)Ws = 15 lbs (weight of skateboard, S)vo = 20 ft/sec (initial speed of runner)v1 = ? (speed of runner & skateboardjust after runner jumps offskateboard)h = 6 ft (desired height R&S wouldreach)Question:Will your friend reach the goal of gliding to a height of at least 6 ft?ApproachDefine the system as the runner and the skateboard. Use conservation of momentum tofind v1, then conservation of energy to find h.Assume that when the runner jumps on the skateboard: (a); the vertical component ofrunner’s momentum is so small it can be ignored, so there is no transfer of momentumand (b) the friction between ground and skateboard is so small it can be neglected, sothere is no transfer of energy as the skateboard and runner move up the slope.DESCRIBE the PHYSICSDiagram(s) and Define QuantitiesMomentumBeforeCollision p i m i v im i m r W rgv i v oMomentum DiagramsMomentumTransferassumeNONE(neglectverticalcomponent ofinitialmomentum)MomentumAfterCollisionp f Mv fM m r m s W r W sgvf v 1 ?Energy beforegoing up slopeE i KE 1 2 Mv i 2v i v 1 ?Energy DiagramsEnergyTransferassumeNONE(neglectfriction)Energy at topof slopeE f PE grav. MgHTarget Quantity: H = ?Quantitative Relationships:p f p i p transfer = 0 and E f E i E transfer = 0

156 Part 3: Teaching a CPS SessionPLAN the SOLUTIONConstruct Specific EquationsFind H: Conservation ofEnergyE f E i E transferMgH 1 2 Mv 1 2 0 H v 1 22gFind v 1 : Conservation ofMomentump f p i p transferMv 1 m r v o 0UnknownsHv 1EXECUTE the PLANCalculate Target Quantity(ies) 125 lbs 2(20 ft/s)H 125+15 lbs2 32 ft/s 2 5.0 ft2So H < 6 ft, so your friend will not reachthe goal of 6 ft above the ground.v 1 m rM v oFind m r /M: Use W=mgm r/Mm rM W r /g(W r W s )/g W rW r W s2 W rv o W So H = r W s 2g2 W H r v o 2W r W s 2gCheck for Units2 [lbs] 2[ft /s] [ft][lbs]+[lbs] [ft/s 2 ]OKEVALUATE the ANSWERIs Answer Properly Stated?Yes. As expected, H has the units offeet.Is Answer Unreasonable?No. If the skateboard were massless,your friend could reach a height ofonly 6.3. So reaching a height of 5feet with a 15 lb skateboard makessense.Is Answer Complete?Yes. As required, we have shown thatyour friend can not reach the goal ofgliding to a height of 6 feet.

Chapter 11: Preparation for a CPS Session 1572. More than 20% Dysfunctional Groups. If appropriate group problems,grading, group structures, and coaching are in place, then at any given time thereshould be no more than 20% dysfunctional groups (i.e., 1 in 5 groups). If younotice an increase in the number of dysfunctional groups in the past few weeks,then one possibility is personality conflicts within groups. In this case, groupsneed the opportunity to resolve these conflicts and decide how to function moreeffectively.Every group evaluation should include the opportunity for students to think aboutand discuss two things: What is something each member did that was helpful for the group? What is something each member could do to make the group even better thenext time they solve a problem together?There are, of course, many ways to do this. If you are changing groups or if you donot have a clear idea why there is an increase in the number of dysfunctional groups,you may decide to use a general form similar to the one shown on page // ofChapter 9. If you have a hypothesis about the cause of the difficulty, you maydecide to ask a few specific questions to test your hypothesis. An example form,shown Figure 10.6, could be adapted for this purpose. The instructor who used thisform suspected that several groups contained students, with strong personalities,who were forcing their groups to follow the plug-and-chug approach ofmanipulating equations rather then the logical and organized approach the instructorwas teaching.Prepare MaterialsThe last step of advance preparation is to make the appropriate number ofphotocopies or prepare a projection of everything you need for your class. Below isa checklist.Problem & Useful Information (one per students), or make a projection of one orboth.Problem Solution: post solution on your web site after class.Answer Sheet (optional) (one per group)Group Functioning Evaluation form (optional) (one per group)Extra copies of Group Roles sheets

158 Part 3: Teaching a CPS SessionFigure 11.6.5. An example of a Group-Functioning Evaluation SheetDate: Group #:Complete the following questions as a team.Low High1. Did all the members of our groupcontribute ideas?1 2 3 4 52. Did all the members of our group listencarefully to the ideas of other groupmembers?1 2 3 4 53. Did we encourage all members tocontribute their ideas? 1 2 3 4 54. What are two specific actions we did today that helped us solve the problem?5. How did each of us contribute to the group's success?6. What is a specific action that would help us do even better next time?Group Signatures:Manager: .Skeptic: .Recorder/Checker: .Summarizer: .

159Chapter 12Teaching a CPS SessionIn this chapter: An outline of instructor actions in teaching a typical CPS session. Description of each instructor action.The previous chapter began with adescription of the three parts of atypical CPS practice session: openingmoves, the middle game, and the end game.The sections below contain a description ofinstructor actions for the opening moves,middle game, and end game. An outline oftheses instructor actions is given in Figure 12.1.Opening Moves Be at the Classroom EarlyThe classroom will probably need some preparation, so it is best to go in and lockthe door, leaving your early students outside. Check out the equipment you willneed to use. Make sure the chairs are set up in appropriate groups. Write the groupassignments and roles on the blackboard if they are new. Include a map of the roomshowing where each group sits. Write instructions for students to sit in their groups. State the Purpose of This CPS Session (~ 2 minutes)(a) Communicate Goal of Session. First tell yourstudents why the problem was selected and what theyshould learn from solving the problem. For example: “Wehave been studying the conservation of energy and theconservation of momentum. Today’s problem illustrateswhen it is useful to apply each conservation law.”(b) Communicate Time Limits. Then tell the students how much time they will haveand what their product should be. For example: “After about 30 minutes, I will

160 Part 3: Teaching a Cooperative Problem Solving SessionFigure 12.1. Outline for Teaching a CPS SessionPreparation Checklist Group/Role assignments (ifnecessary, projected orwritten on board) Projection or photocopies ofProblem & Useful Information(one per person) Photocopies of Answer Sheet (optional)(one per group) Group Evaluation forms (optional one pergroup) and extra photocopies of Group RolesSheet Problem solution for posting on website afterclassOpeningMoves~3-5min.Instructor Actions Be at the classroom early Introduce the problem by telling students:a) what they should learn from solvingproblem;b) the part of the solution you want groupsto put on board Prepare students for group work by:a) showing group/role assignments andclassroom seating map;b) passing out Problem & UsefulInformation and Answer Sheet.What the Students Do• Students sitting and listening• Students move into their groups,and begin to read problem.• Checker/Recorder puts names onanswer sheet.MiddleGame~35 min. Coach groups in problem solving by:a) Monitoring (diagnosing) progress of allgroupsb) Helping groups with the most need. Prepare students for class discussion by:a) giving students a “five-minute warning”b) selecting one person from each group toput specified part of solution on theboard.c) passing out Group Evaluation Sheet(optional)• Solve the problem:- participate in group discussion,- work cooperatively,- check each other’s work.• Finish work on problem• Write part of solution on board• Discuss their group effectivenessEndGame~10 min. Lead a class discussion focusing on whatyou wanted students to learn from solvingthe problem (your goals)a) Start by asking open-ended questions.b) Follow up with questions specific to yourgoal or observed common errors. Discuss group functioning (optional) Post the problem solution after class.• Participate in class discussion

Chapter 12: Teaching a CPS Session 161randomly select one person to write part of their group’s solution on the board.Then we will compare them.” Prepare Students for Group Work (~ 1 minute)(a) Assign or Rotate Roles. If students are working in the same groups, remindthem to rotate roles. If you have assigned new groups, remind students to brieflyintroduce themselves.(b) Problem and Information Sheet. Project of pass out the Problem/information Sheet (oneper group) and an Answer Sheet (one per group). As you do this, make sure all groupsare seated according to your map -- facing each other, close together, but withenough space between groups for you to circulate.Middle Game (~ 30 minutes)There are two instructor actions during the middle game: coaching students inproblem solving, and preparing students for the final class discussion. You willspend most of this time coaching groups. Coach Groups in Problem Solving (~30 minutes)Below is a brief outline of coaching groups. For detailed suggestions of coachingand intervening techniques, see Chapter 13.(a) Diagnose initial difficulties with the problem or group functioning. Once thegroups have settled into their task, spend about five minutes circulating andobserving all groups. Try not to begin coaching until you have observed all groupsat least once. This will allow you to determine if a whole-class intervention isnecessary to clarify the task (e.g., “I noticed that very few groups are drawingappropriate pictures. Be sure to draw the situation at all the important times.”).(b) Monitor groups and intervene to coach whennecessary. Establish a circulation pattern around theroom. Stop and observe each group to see how easilythey are solving the problem and how well they areworking together. Don't spend a long time with any onegroup. Keep well back from students' line of sight sothey don't focus on you. Make a mental note aboutwhich group needs the most help.Intervene and coach the group that needs the most helpto get started. If you spend more than a few minutes with this group, then circulatearound the room again, noting which group needs the most help. Keep repeatingthe cycle of (1) circulate and diagnose, (2) intervene and coach the group that needs

162 Part 3: Teaching a Cooperative Problem Solving SessionFigure 12.2. An example of the parts of a solution each groupcould be asked to write on a wall board.Draw your group’sfree-body force diagramsWrite the equations for thephysics principles and problemconstrains that your group usedto solve the problem.the most help. During this process decide on the part of the problem that would bemost instructive to discuss.(c) Five-minute Warning. About five minutes before you want students to stop,warn the class that they have only five minutes to wind up their solution. Thencirculate around the class once more to determine the progress of the groups. Selectone member of each group to bring part of their solution to the board. In thebeginning of the course, select students who are obviously interested and articulate.Later in the course, it is sometimes effective to occasionally select a student who hasnot participated in their group as much as you would like. This reinforces the factthat all group members need to know and be able to explain what their group did. Prepare Students for Class Discussion (~ 5 minutes)(a) Posting Partial Group Solution. Announce which part of the problem is to beput on the board. An example is shown in Figure 12.2. Those not writing on theboard should compare their solution with those of the other groups going on theboard.(b) Pass out Group Functioning Evaluation form (asneeded). If you decided to have groups evaluate theireffectiveness, pass out the forms (one per group) and havestudents complete the forms while the solution parts aregoing on the board. Give the people who you selected towrite on the board time to participate in the groupevaluation.

Chapter 12: Teaching a CPS Session 163End Game (10 minutes)The end-game discussion should focus on part of the problem that illustrates theprimary goal. The purpose is to help students consolidate their ideas and producediscrepancies that stimulate further thinking and learning.Give students a few minutes to compare the results from each group. Then lead theclass discussion. Lead a Class Discussion (~ 10 minutes)(a) Ask Open-ended Questions. The class discussion is always based around thegroups, with individuals only acting as representatives of a group. This avoidsputting one student "on the spot." Conduct a discussion about the problem solutionwithout (a) telling the students the "right" answer or becoming the final "authority"for the right answers, and (b) without focusing on the "wrong" results of one groupand making them feel stupid or resentful. To avoid these pitfalls, try starting withgeneral, open-ended questions such as: How are the . . . (problem solution part) on the board similar? How do they different?In the beginning of a course, students are usually reticent. They unconsciously playthe “waiting game”. They know that by waiting long enough, the instructor willanswer their own questions and they won’t have to think. We recommend countingsilently up to at least 30 after you have asked a question. Usually students get souncomfortable with the silence that somebody speaks out. If not, call on a group bynumber: “Group 3, what do you think?” Always encourage an individual to gethelp from other group members if he or she is "stuck."(b) Ask Questions Related to Goals or Common Errors. After the general questions,you can become more specific. The questions you ask will depend on what youobserved while groups were solving the problem and what your groups write on theboard.Encourage groups to talk to each other byredirecting the discussion back to the groups.For example, when a group reports theiranswer to a question, ask the rest of the class tocomment: "What do the rest of you thinkabout that?" This helps avoid the problem ofyou becoming the final "authority" for the rightanswer. Encourage students to go to theboard, point out something a group wrote, andask the groups questions.

164 Part 3: Teaching a Cooperative Problem Solving Session Discuss group functioning (optional, ~ 5 minutes)An occasional class discussion of group functioning is essential. Students need tohear the difficulties other groups are having, discuss different ways to solve thesedifficulties, and receive feedback from you (see Chapter 9, pages // and //).Randomly call on one member of from each group to report their group’s answer tothe following question on the Evaluation form: one difficulty they encountered working together, or one way they could interact better next time.After each answer, ask the class for additional suggestions about ways to handle thedifficulties. Then add your own feedback from observing your groups (e.g., "Inoticed that many groups are coming to an agreement too quickly, withoutconsidering all the possibilities. What might you do in your groups to avoid this?") Post the problem solution.Posting the solution on your website is important to students. They need to seegood examples of solutions to improve their own problem solving skills. Again, it isimportant to post them as the last thing you do – just after students have left theclassroom. If you post the solution earlier, your students will ignore the classsession.

165Chapter 13Coaching StudentsDuring Group WorkIn this chapter: How to monitor groups and diagnose their difficulties. How to intervene and coach dysfunctional groups. How to intervene and coach groups having difficulty applying physics concepts and principlesto solve a problem.There are two important instructor actions involved in efficient and timelycoaching of groups while they are working to solve a problem: monitoring all groups and diagnosing their difficulties; and intervening and coaching the groups that need the most help.Coaching groups that are solving problems is similar to triage in a medicalemergency room. When there are more patients than available doctors, doctors firstdiagnose what is wrong with each patient to decide which patients need immediatecare and which can wait a short time. The doctors then treat the patient with themost need first, then the second patient, and so on. Similarly, with CPS theinstructor needs to first diagnose the “state of health” of each group by observingand listening to each group (without interacting with the groups). With CPS, youdiagnose: what physics concepts and problem-solving procedures each group does anddoes not understand; and what difficulties group members are having working together cooperatively.As with medical triage, your next step is to intervene with the group that is havingthe most difficulty with the physics or with group functioning.This chapter contains recommendations of how to monitor and coach groups.

166 Part 3: Teaching a Cooperative Problem Solving SessionMonitor and DiagnoseThe following steps are helpful to monitor and diagnose the progress of all groups:Step 1. Establish a circulation patternaround the room. Stop and observe eachgroup to see how easily they are solving theproblem and how well they are workingtogether. Don't spend a long timeobserving any one group. Keep well backfrom students' line of sight so they don'tfocus on you.Step 2. Make mental notes about eachgroup’s difficulty, if any, with groupfunctioning or with applying physicsprinciples to the problem solution, so you know which group to return to first.Step 3. If several groups are having thesame difficulty, you may want to stop thewhole class and clarify the task or makeshort, additional comments that will help thestudents get back on track. For example,there is a tendency for students toimmediately try to plug numbers intoequations each time new physics principlesare introduced. If about half of your groupsare doing this, stop the whole class. Remindyour students that the first step in problemsolving is a thorough analysis of the problembefore the generation of mathematicalequations.If you begin interveningtoo soon (without firstdiagnosing all groups), itis not fair to the lastgroups. By the time yourecognize that all groupsmay have the samedifficulty, the last groupswill have wastedconsiderable time.Intervene and CoachFrom your observations, decide if any group is struggling and needs attentionurgently. Return to that group and watch for a few minutes to diagnose the exactnature of the problem, then join the group at eye level. You could kneel down orsit on a chair, but try not to loom over the students.If you spend more than a few minutes with this group, circulate around the roomagain, noting which other groups need your help. Keep repeating the cycle ofa) circulate and diagnose, (b) intervene with the group that needs the most help.

Chapter 13: Coaching Students during CPS 167In well functioning groups, members share the roles of manager, checker, explainer, andskeptic, and role assumption usually fluctuates over the time students are solving aproblem. Students in these groups do not need to be reminded to "stick to theirroles." For dysfunctional groups, however, assigned group roles are an importantpart of intervention strategies. For example, one way to intervene with adysfunctional group (e.g., a dominant student, one person working alone) is to ask:"Who is the manager (or skeptic/summarizer, or recorder/checker, depending onthe dysfunctional group behavior)? What should you be doing to help resolve thisproblem?" If the student does not have any suggestions, then model severalpossibilities.Coaching Dysfunctional GroupsFirst Few CPS Sessions.In the first CPS sessions, students with no prior experience with cooperativelearning often do not understand their role in a group co-construction of a problemsolution. Students do not organize their own thoughts about a problem beforelaunching into an algorithmic process. Naturally they are more comfortablecomparing answers or equations than sharing their thought process about thephysics. Without coaching, there is a tendency to solve problems individually, andcompare progress to determine the “best” solution. The three examples belowillustrate some common difficulties and possible interventions for the first CPSsessions.Example 1: Individual Problem Solving.You observe a group in which the membersare not talking to each other, but solving thegroup problem individually.Say something like: “I notice that you aresolving the problem individually, not as agroup. Who is the Recorder/Checker? Youshould be the only person writing thesolution. Manager and Skeptic/Summarizer,put your pencils away and work with theRecorder/Checker to solve the problem.”

168 Part 3: Teaching a Cooperative Problem Solving SessionIf necessary, make the students rearrange their chairs so they can all see what theRecorder/Checker is writing. If the students persist in solving the problemindividually and only then return to the group to compare answers, explain againthat they should be solving the problem together. Take the pencils from theManager and Skeptic (return them at the end of class), and have the group read theGroup Role sheet again. Do not leave until they have started solving the problemtogether.Example 2: A Lone Problem Solver. You observe a group in which twomembers, including the Recorder/Checker, are working together, but one member,the “loner”, is working alone to solve the problem. First, try to determine why theloner is solving the problem alone. Say something like: “I notice that while two ofyou are working together, you (loner) appear to be solving the problem by yourself.What are each of your group roles? Why are you (loner), as the group Manager (orSkeptic/Summarizer) solving the problem by yourself?”Frequently, the loner will sheepishlymumble something about not beingused to working in a group. Thisindividual may need only a gentlereminder to give group work a try.Ask the Recorder/Checker to explainto the loner what they have done sofar to solve the problem. If necessary,make the students rearrange theirchairs so they can all see what theRecorder/Checker is writing.Occasionally, a loner is more adamant about needing to solve the problem alonebefore talking with the group. Maintain a sympathetic attitude, but explain to theloner that the research shows that all students learn much more about physics andproblem-solving procedures when they construct problem solutions together, whichis why you are having them do it. Although it may seem difficult at first, insist thatthe loner try it. Tell the individual to put their pencil away and ask theRecorder/Checker to explain to the loner what they have done so far to solve theproblem.Example 3: A Non-participant. You observe a group in which one member doesnot appear to be engaged in the group problem-solving process.

Chapter 13: Coaching Students during CPS 169Try to determine why the student appears to be disengaged. For example, if thestudents are sitting in a row and not facing each other, have the students to get upand rearrange the chairs so they sit facing each other. Ask the student to explainwhat the group is doing and why. [This emphasizes the fact that all group membersneed to be able to explain each step in solving a problem.] If the student candescribe what the group is doing and why, then they may be a quiet student whopays attention, but does not speak as often as the others. You do not need tointervene further.If the student does not have a clearidea of what the other groupmembers are doing, they may bewhat is called a “free-rider” -- aperson who leaves it to others tosolve the problem. Ask the freerider: What is your group role?What should you be doing to helpyour group solve this problem?” [Ifnecessary, have the free rider readthe role description from the Group Role sheet.] If the free-rider is not theManager, ask the Manager what could be done to make sure everyone, including thefree-rider, participates in solving the problem.Later CPS Sessions.With appropriate structure (see Chapters 7 through 9) and coaching, most studentslearn to function in groups relatively well. Occasionally, however, a group mayexhibit one of the following dysfunctional behaviors 1 : Lower-achievement members sometimes "leave it to John" to solve the groupproblem, creating a free-rider effect. At the same time, higher achievinggroup members may expend decreasing amounts of effort because they feelexploited by the others, the sucker effect. This sucker effect is unusual whengroup problems are graded occasionally. Higher-performance group members may be deferred to and take overleadership roles in ways that benefit them at the expense of the other groupmembers (the dominant student or the rich-get-richer effect). Groups with no natural leaders may avoid conflict by "voting" or not makingany decision rather than discussing an issue (conflict avoidance effect). Group members argue vehemently for their point of view and are unable tolisten to each other or come to a group consensus (destructive conflicteffect).The last section included an example of how to intervene in a group with a “freerider.”The two examples below suggest how to coach groups with a dominantstudents or a conflict.

170 Part 3: Teaching a Cooperative Problem Solving SessionExample 1: Dominant Student. You observe a group in which one member isdoing almost all of the talking, while the other members appear somewhatdisengaged and lethargic.In this case, all members are failing in their roles. First tell the group: “I notice thatone person appears to be doing all the talking in this group.“ Then ask: Manager,what could you be doing to make sure that all members of your group contributetheir ideas?” If the manager has no ideas, then either have the group read theirGroup Role sheet (early in course) or make a suggestion, such as: “For each step inyour problem solving process, ask each member of your group what they think.”Point to a specific part of the group’s solution and model some specific questionsthe Manager could ask.Repeat this procedure with each group member. Ask: “Checker/Recorder, whatcould you be doing to make sure that all members understand and can explaineverything that is written down?” [Periodically ask each member if they understandand agree with everything written down. Point to part of the group’s solution andmodel some specific questions.] Ask: “Skeptic, what could you be doing to makesure that alternative ideas are being considered by the group?” [Be sure to ask for ajustification for an idea, and suggest alternative ideas. Point to specific parts of thegroup’s solution and model specific questions the skeptic could ask.]Example 2: Conflict Avoidanceor Destructive Conflict. Youobserve a group that cannot seemto reach a decision, but does notappear to have any strategy thatleads to convergence (conflictavoidance) or a group that isarguing loudly, but does not appearto be resolving their conflict(destructive conflict). Ask thegroup: "Who is theSkeptic/Summarizer (or Summarizer in a four-member group)? I noticed that youare having difficulty deciding . . . . . Summarizer, what could you be doing to helpthe group come to a decision that is agreeable to all of you? If the Summarizer hasno idea, then either have the group read the Group Role sheet again (early in course)or give some suggestions, such as: “Stop and summarize your different ideas. Thendiscuss the merits of each idea. For example, you could . . . ." The specificsuggestions you give will depend on the exact nature of the decision.

Chapter 13: Coaching Students during CPS 171Coaching Groups with Physics DifficultiesAs the number of dysfunctional groups decreases, you will spend more of your timecoaching groups that are having difficulty applying physics concepts and principlesto solve the problem. The general approach to coaching is to give a group justenough help to get them back on track, then leave. That is, spend as little time aspossible with a group, then go to the next group that needs help, and so on. Beloware some general guidelines for coaching groups with physics difficulties.Step 1. Before you intervene, listen to the discussion in a group for a few minutesand look at what the checker/recorder is drawing and writing. Diagnose the group’sspecific difficulty. The checklist for grading feedback (Chapter 9, page //) can beuseful for this purpose.Step 2. Based on the nature of the group’s difficulty, decide how to begin yourcoaching of the group. There are two general coaching approaches, depending onwhether you can point to the difficulty on the group’s answer sheet. Use Group Roles. Point to something on the answer sheet and state thegeneral nature of the difficulty or error. Then ask: “Then ask: "Who is themanager (or skeptic/summarizer, or recorder/checker)? What could you bedoing to help resolve this difficulty?" If the student/group does not have anysuggestions, then model several possibilities. General Questions. If you can not point to something specific written onthe group’s answer sheet, begin by asking the group some general questionsto find out what they are thinking, such as: (a) What are you doing? (b) Whyare you doing it? and (c) How will that help you?Step 3. Based on the answers you get to your initial question(s), ask additionalquestions until you get the group thinking about how to correct their difficulty. Thatis, try to give a group just enough help to get them back on track, then leave. Checkback with the group later to see if your coaching was sufficient for the group todiscuss the difficulty and get back on track.Examples of Using Group Roles in CoachingSuppose your students are solving amodified Atwood machine problem, asshown in the diagram at right. As part ofthe solution, students must find thetension of the rope. Below are someexamples of a coaching technique thatuses group roles.W caW h

172 Part 3: Teaching a Cooperative Problem Solving SessionExample 1: Misunderstanding of PhysicsConcept. You observe that a group has drawn thefrictional force in the wrong direction on theirdiagram. Point to the diagram: “There issomething wrong with one of the forces in thisdiagram. Skeptic, what questions could you askabout each of these forces?” When the group hasresponded (i.e., Does each interaction result in apush or a pull on the carton? In what direction?),then leave the group.W cTNf kExample 2: Improper Construction of aSpecific Equation. You observe that a group hasdrawn a correct force diagram, but there is anincorrect sign for the frictional force in their 2ndLaw component equation:Tf kFx maxWT W sin W cos cc c agPoint to the force diagram and the equation: “I think you made a mistake intranslating from your diagram to this equation. Skeptic, what questions could youask about each translation?” When the group has responded, then leave the group.Example 3: Diagram Missing. You observe that a group has not drawn aseparate force diagram. Their 2nd Law equation is correct except for the wrong signfor the frictional force. Point to the equation: “I think there is a mistake in thisequation. Manager, what is an important part of analyzing a problem that couldprevent a mistake in this equation?” When the group has responded with “a forcediagram,” leave the group. It they don’t, you should suggest they draw one.W cNExample 4: Major Misconception. Youobserve a group that has not drawn separateforce diagrams for the carton and thehanging weight. Instead, they sketchedsome forces on the picture, as shown atright. In addition, they did not start theirequations with Newton’s Second Law in itsgeneral form, F x =ma x . Instead, the firstequation is:T W h f k W c sin W h W c cos W c sin f k W cEquations of this type often indicate a misconception about Newton’s 2nd Law. Wehave found that about 20% of students in the calculus-based course solve Newton’sLaw problems by setting the unknown force (tension in this problem) equal to theaNW c sinW h

Chapter 13: Coaching Students during CPS 173sum of the known forces, in this case all the other forces acting on the carton andthe hanging weight (see Chapter 14, pages //-//). [In addition, about 20% ofstudents in a traditional class solve Newton’s Law problems by setting the unknownforce (e.g., tension) equal to “ma,” or by setting the sum of the forces equal to zeroeven when there is an acceleration.]Point to the equation: “I don’t understand this equation. Checker/recorder, couldyou describe how your group arrived at this equation?” Specific follow-up questionswill depend on the response of the group. If you have Newton’s Second Law(F x =ma x ) on the Problem & Information sheet, then you could point to thisequation and ask the group what this equation means. Finally, you may need tocoach the group through drawing free-body force diagrams for each object (cartonand hanging weight).General-Questions Coaching TechniqueSometimes, it is impossible to identify a specific error even though the solution pathis obviously incorrect. By the time you get to a group, they may have severalinterrelated difficulties. Your intervention with this group will take longer. You canstart coaching by asking the group: (a) What are you doing? (b) Why are you doingit? and (c) How will that help you? This often provides you with enoughinformation to diagnose the problems and deal with them one at a time. Always tryto ask questions, rather than give answers.

174 Part 3: Teaching a Cooperative Problem Solving Session

Part175

176In this part . . .This part of the book presents some advanced techniques in CooperativeProblem Solving (CPS). It assumes that minimally you have read or know thecontent of Chapters 2, 3, 4, 5, 7, and 11, and have tried some cooperativeproblem solving with your classes, using context-rich problems formAppendices B and C or from our on-line archive of context-rich problems.Chapter 14 describes how to personalize a problem-solving framework tomatch your students and your goals and approach to your introductory course.It provides examples of different research-based, problem-solving frameworksthat you can modify to fit your own needs.Chapter 15 describes a procedure for constructing your own context-richproblems from textbook exercises or end-of-chapter problems. A review ofthe features of context-rich problems is followed by an explanation of theprocedure, and examples are provided for your practice in writing your owncontext-rich problems.Chapter 16 provides guidance on how to judge the suitability of a context-richproblem for use by individuals or groups in either a practice or exam situation.Twenty-one traits are described that make a more difficult to solve, and achecklist is provided for determining the difficulty level of a problem and it’ssuitability for an intended use.

177Chapter 14Personalizing a ProblemsolvingFrameworkIn this chapter: Five steps for personalizing a problem-solving framework to fit your students and yourapproach to your introductory course. Examples of research-based, problem-solving frameworks. Examples of problem-solving frameworks in textbooks.In Chapters 3 and 4 we described the problem-solving framework that we use inour introductory physics courses. A problem-solving framework is a logicaland organized guide to help students arrive at a solution to real problems. Itgets students started, guides them to what to consider, organizes their mathematics,and helps them determine if their answer is reasonable.The problem-solving framework presented in Chapter 4 and expanded in Chapter 5is one several that have been used for introductory physics. All research-basedframeworks are specific adaptations of the general framework used by experts in allfields (see Chapter 4, page //). Although these frameworks are very similar, theywere developed and tested for different populations of students. They divideimportant problem-solving actions into a different number of steps and sub-steps,describe the same actions in different ways and emphasize different heuristicsdepending on the backgrounds and needs of population of students for whom theywere developed.The details of the framework you decide to teach should be tailored to the needsand backgrounds of your students and to your approach to the course you areteaching. In this chapter we describe guidelines to help you personalize aframework for your students. There are five major steps in this procedure, outlinedbelow.A problem-solving framework is only a guide for students -- actions for students toconsider as they solve problems. It does not present a set of linear steps to be

178 Part 4: Personalize a Problem Solving Framework and Problemsfollowed. Problem solving usually involves looping back and forth both withinsteps and between steps.Figure 14.1. Checklist of Symptoms of Students’ Difficulties Solving Problems 1. Difficulty visualizing a physical situation. Symptoms include physicallyimpossible results. No pictures or diagrams are drawn. 2. Disconnect between physics and reality. Symptoms include physically impossibleresults; excessive emphasis on the exact meaning of words (the “lawyer” approach toa problem); difficulty in applying knowledge to slightly different situations; anddifficulty applying knowledge consistently within a single situation or across similarsituations. 3. No recognition of common physics concepts. Major symptom is difficulty inapplying knowledge to slightly different situations. 4. Difficulty applying of fundamental concepts consistently. Symptoms includedifficulty in applying knowledge to slightly different situations; and incorrectalternative conceptions remain. 5. Lack of a coherent conceptual framework. Symptom is that manymisconceptions remain. Students have difficulty applying their knowledge to slightlydifferent situations and difficulty applying knowledge consistently, even within asingle situation. 6. Lack of a logical analysis. Symptoms include random equations and/or theinability to begin a solution. 7. Over-reliance on pattern matching. Symptom is students solving the “wrong”problem either through over-simplification or misreading of the problem. 8. Lack of mathematical rigor. Symptoms include frequent “algebraic mistakes” andmathematical “magic” in solutions (e.g., m 1 = m 2 = 1).Step . Clarify the situation. Analyze your students’ solutions from homeworkand/or exams to determine your students’ problem solving difficulties.Step . Gather Additional Information. Determine how your students’difficulties have been addressed in some existing research-based,problem-solving frameworks.Step . Construct Your Own Framework. Build a problem-solving frameworkfor your students by adapting research-based frameworks.Step . Reconcile Your Framework with Your Textbook Strategy. Figure outhow you can use your framework in conjunction with the problemsolvingstrategies in your textbook.Step . Evaluate Your Framework. Use your problem-solving framework with aclass. Based on the results, modify your framework as necessary byrepeating the above steps.

Chapter 14: Personalizing a Problem Solving Framework 179Each section of this chapter describes the specific actions in each of these steps, andprovides some examples.Step . Clarify the SituationThe first step in personalizing a problem-solving framework is to analyze yourstudents’ written solutions to practice and exam problems to determine theproblem-solving difficulties of your students. Your experience in working withstudents during your office hours or other discussions will also help. The Checklist ofSymptoms of Student Difficulties in Figure 14.1 and the table in Chapter 3, page //,might be helpful. Note that a symptom may have more than one cause.It is tempting, at first glance, to ascribe many student difficulties to mathematicalweaknesses because students’ solutions tend to be mostly mathematics.Considerable research indicates, however, that mathematics is only a minor elementof students’ difficulties solving real problems. i For example, Figures 14.2 – 14.4 givethe results of our analysis of student solutions to three final examination problemsfor a calculus-based physics course (two problems from the first semester and onefrom the second semester). These results were used to develop our problem-solvingframework for that course.The problems were selected because they required students to apply their knowledgeto slightly different situations than they had encountered in their course andtextbook. They were also standard problems for a calculus-based course. We beganby analyzing the students’ solutions to the Modified Atwood-machine Problem (Figure14.2). At first, the major student errors seemed to in the mathematics. However,we adopted the following procedure:Look first at the student’s diagrams (if any).Then look at the first equation written down (or the first equation in eachsub-part). Does the equation match the diagram? Is the equation anapplication of a fundamental concept or principle, or something else?Based on this procedure, we classified over 250 studentsolutions to each problem for major errors (i.e., the error thatprevented students from arriving at a correct solution). iiApproximately 40% of the students solved the three finalexamination problems correctly (or with minor errors). Asexpected, only about 10% of our students had majordifficulties that could be directly ascribed to mathematics, asshown in Figure 14.2 and Figure 14.4.In contrast, approximately 50% of our students failed to solve each problem becauseof incorrect approaches to the problems. These students either drew no pictures ordiagrams or drew incomplete diagrams. This is a symptom of the problem-solvingdifficulty of visualizing a physical situation (Difficulty 1 in Figure 14.1). Approximatelyone-half of our students could not apply their knowledge consistently (primarilyDifficulties 3 and 4) . They did not appear to have integrated their knowledge into acoherent conceptual framework (Difficulty 5 in Figure 14.1).

180 Part 4: Personalize a Problem Solving Framework and ProblemsFigure 14.2. Major errors in students’ solutions to Modified Atwood-machine ProblemModified Atwood-machine Problem. In the diagram shown atright, block 1 of mass 1.5 kg and block 2 of mass 4 kg areconnected by a light taut rope that passes over a frictionlesspulley. Block 2 is just over the edge of the ramp inclined at anangel of 30 o , and the blocks have a coefficient of sliding friction of0.21 with the surface. At time t = 0, the system is given an initialspeed of 11 m/s that starts block 2 down the ramp. Find thetension in the rope.Major Type of Error in Students’ Solutions%(N=272)1. Correct or minor errors Total: 292. Careless, many omissions, no sense of order 93. Incorrect Physics Approaches52a. Funknown = ma. The unknown force, in this case the tension in therope, is mass times acceleration (ma). Usually no force diagram (or veryminimal diagram) is drawn. These students appear to have no idea whatthe “tension” force is.6T = F = ma, where the acceleration could be g, gsin, or v ob. Funknown = Fknown. The unknown force (tension) is the sum of allthe known forces acting on the two blocks. Usually only a minimal forcediagram is drawn, often without the tension force.22T = F 1 + F 2 - F 3 . . .c. Tension = Friction. The unknown force (tension) is the frictional forceon m1 or the sum of the frictional forces on m1 and m2. Usuallyminimal force diagrams are drawn, often without the tension forceshown. (They may be setting F = 0 in their heads).11T = m 1 gord. Incomplete, can’t tell.T = f 1 +f 2 = m 1 g + m 2 gcos134. Mathematical Difficulties9a. Can’t solve simultaneous equationsb. Trigonometry or algebra errors63

Chapter 14: Personalizing a Problem Solving Framework 181Figure 14.3. Incorrect physics approaches in students’ solutionsto the Enterprise ProblemEnterprise Problem. The “Enterprise, a ride at the Valley Fair Amusement Park, consists of avertical wheel if radius 9 meters rotating about a fixed horizontal axis with seats for theoccupants around its outer edge. The wheel rotates so that the occupants are moving at 11m/s. The seats pivot so the occupants’ heads are towards the center of the wheel. When a 56-kg woman is upside down at the top of the wheel, what is the force she exerts on the seat?%(N=289)Incorrect Physics Approaches in Students’ Solutions46a. Funknown = ma. The unknown force (in this case the force of thewoman on the seat, usually written as “F”) is mass times acceleration(ma). Usually no force diagram is drawn.9b. Funknown = Fknown. The unknown force (force of the woman on theseat) is the sum of all the known forces in the problem. The force of theseat on the woman is not drawn (or given a symbol). F C or mv 2 /r isdrawn as a force vector.18c. F = 0. The sum of the forces acting on the object (woman) is zero. Inthis case, the unknown force (force of the seat on the woman) is shownin the force diagram. F C or mv 2 /r is drawn as a force vector.11d. N in wrong direction. The normal force is drawn in wrong direction,but the application of Newton’s second law is correct.8

182 Part 4: Personalize a Problem Solving Framework and ProblemsFigure 14.4. Major errors in students’ solutions to the Wave ProblemWave Problem. A violin String 55 cm in length and placed near a loudspeaker is observed torespond strongly when the speaker is driven at a frequency of 1320 Hz exhibiting two nodesbetween endpoints. What is the tension in the spring if it has a mass of 0.5 grams?Major Type of Error in Students’ Solutions1. Correct (T = v 2 , v = f)a. No mistakesb. Math Mistake (units, calculator, forgot to square v)%(N=217)443592. Incorrect Approaches 49Incorrect Approach for Determining String Density ( = m or = mg)Incorrect Approach for Determining Wave Velocity (v).(No useful diagram drawn for determining )a. Used v = 2Lf n /n, with n = # of nodes.b. Used formula for constructive interference ∆L = nc. Used velocity of soundd. Used other relationships (pendulum, harmonic oscillator)e. No idea how to calculate v, so quit.445131215143. Incorrect Approach for Determining T 54. Nothing or little written 3Student solutions to both mechanics problems (the Modified Atwood-machine Problemand Enterprise Problem) indicated similar incorrect approaches: F unknown = ma. The unknown force is always the mass times theacceleration (ma). F unknown = F known . The unknown force is always the sum of all the knownforces in the problem. F = 0. The sum of the forces acting on an object is always zero.These three approaches could indicate students may have misconceptions aboutacceleration, the nature of forces, or about the meaning of Newton’s second law(Difficulties 4 and 5 in Figure 14.1). This view is supported by our results forwritten conceptual questions (see Chapter 10, pages //-//). At the end of thequarter on mechanics, 57% of students still confused acceleration and velocity, 39%did not understand the nature of forces (e.g., included the force acceleration or theforce of momentum), and 75% did not understand that acceleration is caused by thesum of the forces acting on a object.

Chapter 14: Personalizing a Problem Solving Framework 183The three incorrect approaches could also be due to the pattern-matching strategyfor solving quantitative problems – memorizing the series of equations needed tosolve different types of problems (Difficulties 5 and 7). That is, students use adifferent network of knowledge to answer conceptual questions than to answerquantitative questions. This view is supported in a study by Mel Sabello and EdwardRedish. iii When administering a problem with several parts, they found that 17% ofstudents set the sum of the forces equal to zero when solving the quantitative partof a problem, despite the fact that they drew a non-zero acceleration vector in anearlier qualitative part of the problem. This indicates a clear disconnect betweentheir qualitative knowledge and their quantitative knowledge. The reality is probablysomewhere between these two extremes, with students having misconceptions andexhibiting an over reliance on pattern matching.Additional analysis of similar problems and open-ended conceptual problemsindicated that the primary difference between the students in the algebra-based andcalculus-based courses was their mathematical dexterity. About 90% of ourcalculus-based students had sufficient mathematical dexterity that, together withtheir over-reliance on pattern matching, they could arrive at a numerical answer. Inour algebra-based course, students made more mathematical mistakes, somestudents would give up half-way through the algebra, and more students used thenovice plug-and-chug strategy – randomly plugging numbers into memorizedformulas until they arrive at a numerical answer (Chapter 2, pages // - //).Pattern-matching Novice StrategyPlug-and-chug Novice StrategyOur initial problem-solving framework was based on the observation that themajority of our students in both the algebra-based or the calculus-based physicscourse exhibited the problem-solving difficulties in Figure 14.1 on the finalexamination. Subsequent analysis of students’ problem solutions led to therefinement and elaboration of slightly different frameworks for each course, asdiscussed in Chapter 5.Step . Gather Additional InformationOnce you have determined your students’ problem-solving difficulties, the next stepis to determine these difficulties have been addressed in some existing researchbased,problem-solving frameworks. There are several research-based frameworks

184 Part 4: Personalize a Problem Solving Framework and ProblemsFigure 14.5. Three examples of research-based problem-solving frameworks.PHYSICSFRAMEWORKbyFred Reif (1996)GENERALFRAMEWORKbyGeorge Polya (1957)PHYSICSFRAMEWORKbyHeller & Heller (1992)1. ANALYZE THEPROBLEM1. UNDERSTANDTHE PROBLEM1. FOCUS ONTHE PROBLEM2. DESCRIBETHE PHYSICS2. PLAN ASOLUTION3. PLAN ASOLUTION2. CONSTRUCT ASOLUTION3. CARRY OUT THEPLAN4. EXECUTETHE PLAN3. CHECK ANDREVISE4. LOOKBACK5. EVALUATETHE SOLUTIONfor physics that have been developed and successfully used, iv , v , vi all based on thegeneral framework developed by George Polya, vii Two examples are outlined inFigure 14.5. Each framework divides the important actions into a different numberof steps and sub-steps, describes the same actions in different ways, and emphasizesdifferent heuristics depending on the backgrounds and needs of population ofstudents for whom they were developed.For each framework consider: the specific actions in each step; how these actions are carried out in solving a physics problem; the specific problem-solving difficulties each step and/or action is designedto help students overcome; the match with the problem-solving difficulties your students exhibit.Step . Construct Your Problem-solving FrameworkBegin with a Research-based Framework. Choose the research-based, problemsolvingframework that best addresses the major difficulties of the majority of yourstudents.

Chapter 14: Personalizing a Problem Solving Framework 185Figure 14.6. Outline of the Competent Problem-solving Framework: Calculus Version1. Focus the Problem. Establish a clear mental image of the problem.A. Visualize the situation and events by sketching a useful picture.• Show how the objects are related spatially, Show the time sequence of events, especiallythe initial and final states of the objects. Add times to the drawing when an objectexperiences an abrupt change in interaction.• Write down the known information, giving each quantity a symbolic name and addingthat information to the picture.B. Precisely state the question to be answered in terms you can calculate.C. Identify physics approach(es) that might be useful to reach a solution.• Which fundamental principle(s) of physics (e.g., kinematics, Newton’s Laws, conservationof energy) might be useful this problem situation.• List any approximations (e.g., assume kinetic friction is negligible) or problemconstraints (e.g., constant acceleration, T 1 = T 2 , uniform electric field) that arereasonable in this situation.2. Describe the PhysicsA. Draw any necessary diagrams (e.g., motion diagram, force diagram, momentum diagram,energy table) with coordinate systems that are consistent with your approach(es).• Define consistent and unique symbols for any quantities that are relevant to thesituation.• Identify which of these quantities is known and which is unknown.B. Identify the target quantity(s) that will provide the answer to the question.C. Assemble the appropriate equations that mathematically give the physics principles,approximations, and problem constraints identified in your approach.3. Plan a SolutionA. Construct a logical chain of equations from those identified in the previous step, leadingfrom the target quantity to quantities that are known.• Choose an equation that contains the target quantity and write it down. Identify otherunknowns in that equation.• Choose a new equation for one of these unknowns. Write down this equation and notethe unknown quantity this equation was chosen to determine.• Continue this process for each unknown.B. Determine if this chain of equations is sufficient to solve for the target quantity bycomparing the number of unknown quantities to the number of equations.4. Execute the PlanA. Follow the outline from in the previous step.• Arrive at an algebraic equation for your target quantity by following your chain ofequations in reverse from the order you constructed your plan.• Check the units of your algebraic equation before putting in numbers.• Use numerical values to calculate the target quantity.5. Evaluate the AnswerA. Does the mathematical result answer the question asked with appropriate units?B. Is the result unreasonable?

186 Part 4: Personalize a Problem Solving Framework and ProblemsFigure 14.7. The problem-solving framework by Fred Reiffor a calculus-based introductory course in mechanics.1. Analyze the Problem: Bring the problem into a form facilitating its subsequent solution.A. Basic Description—clearly specify the problem by• describing the situation, summarizing by drawing diagram(s) accompanied by somewords, and by introducing useful symbols; and• specifying compactly the goal(s) of the problem (wanted unknowns, symbolically ornumerically)B. Refined Description—analyze the problem further by• specifying the time-sequence of events (e.g., by visualizing the motion of objects as theymight be observed in successive movie frames, and identifying the time intervals wherethe description of the situation is distinctly different (e.g., where acceleration of objectis different); and• describing the situation in terms of important physics concepts (e.g., by specifyinginformation about velocity, acceleration, forces, etc.).2. Construct a Solution: Solve simpler sub-problems repeatedly until the original problem has beensolved.A. Choose sub-problems by:• examining the status of the problem at any stage by identifying the available knownand unknown information, and the obstacles hindering a solution;• identifying available options for sub-problems that can help overcome the obstacles;• selecting a useful sub-problem among these options.B. If the obstacle is lack of useful information, then apply a basic relation (from generalphysics knowledge, such as F net = ma, f k = N, x = 1/2 a x t 2 ) to some object or system atsome time (or between some times) along some direction.C. When an available useful relation contains an unwanted unknown, eliminate theunwanted quantity by combining two (or more) relations containing this quantity.Note: Keep track of wanted unknowns (underlined twice) and unwanted unknowns (underlinedonce).3. Check and Revise: A solution is rarely free of errors and should be regarded as provisionaluntil checked and appropriately revised.A. Goals Attained? Has all wanted information been found?B. Well-specified? Are answers expressed in terms of known quantities? Are units specified?Are both magnitudes and directions of vectors specified?C. Self-consistent? Are units in equations consistent? Are signs (or directions) on both sidesof an equation consistent?D. Consistent with other known information? Are values sensible (e.g., consistent withknown magnitudes)? Are answers consistent with special cases (e.g., with extreme orspecially simple cases)? Are answers consistent with known dependence (e.g., withknowledge of how quantities increase or decrease)?E. Optimal? Are answers and solution as clear and simple as possible? Is answer a generalalgebraic expression rather than a mere number?

Chapter 14: Personalizing a Problem Solving Framework 187Language. Next consider how your students would interpret language calling forthe specific actions in the framework. Since the framework was developed as aspecific implementation of Polya’s general framework used by experts in all fields, itwill probably need to be modified for your particular population of students.1. Retain the elements of each research-based framework that:• seem directed at your students’ most common problem-solvingdifficulties; and• are consistent with your course goals and approach.2. Reduce or eliminate steps in the framework that address difficulties notexhibited by your students.3. Make sure that the framework describes a complete, logical problem-solvingprocedure from your point of view. Fill in steps if necessary to assurecompleteness.4. Choose language to describe the steps and actions of your modified problemsolvingframework that is most meaningful to you and your students (see alsoStep ).Compare with Research-Based Frameworks. Make surethat all of the features of a research-based frameworkare incorporated (see Chapter 4, pages //-//).Check Your Framework. Check that the framework willbe useful in all parts of your course by solving problemsusing its procedure for different physics topics.Remember, a problem-solving framework is most useful at the beginning of thecourse. When students become more familiar with the framework and comfortableusing the framework, the different steps begin to merge. For example, towards theend of the first semester, our students in the calculus-based course merge Steps 1and 2 into one description, and merge Steps 3 and 4 into one procedure. This is anexample of our 2nd Law of Instruction: Don’t change course in midstream; structure earlythen gradually reduce the structureExample 1. Compare Algebra versus Calculus VersionsFigure 14.6 outlines our research-based Competent Problem-solving Framework forstudents in our calculus-based course. viii Compare this framework with the outlineof our framework for students in our algebra-based course (Chapter 4, page //). ixSteps 1, 2, 4, and 5 in these frameworks are identical because we found that studentsin both our algebra-based and calculus-based courses had similar conceptualdifficulties. These difficulties, addressed in steps 1, 2, and 5, were in visualizing aphysical situation, connecting physics to reality, recognizing a common physicstheme, applying fundamental concepts, and integrating knowledge into a coherentconceptual framework.

188 Part 4: Personalize a Problem Solving Framework and ProblemsIn addition almost all students had difficulty solving problems in a logical, organizedprogression. These difficulties were addressed in steps 3 and 4 of the framework.The actions in step 3 are slightly different because the calculus-based students havemore dexterity in mathematics than the algebra-based students. Most of thecalculus-based students (but not all) could follow their plan without needing to writean outline of their actions. The algebra-based students, on the other hand, neededthis organizational tool to keep from becoming consumed by superfluousmathematical manipulation.Example 2. Compare Two Calculus-based VersionsFigure 14.7 contains an outline of another research-based framework developed byFred Reif and his group at Carnegie Mellon University 3 for use in the first semester(mechanics) calculus-based course. In contrast, our Competent framework (Figure14.6) is generalized for use in both semesters of an introductory course. Reif’sframework breaks the generalized procedure into three major steps, whereas ourCompetent framework has five major steps (see Figure 14.5).Fred Reif’s FrameworkOur Competent FrameworkOne major difference between the two frameworks is in the heuristic emphasized toplan and construct a solution. A heuristic is a rule of thumb – a procedure that isboth powerful and general, but not absolutely guaranteed to work. (see Chapter 4,pages // - //). Reif’s framework emphasizes the heuristic of breaking a probleminto sub-problems that you can solve (Step 2). Our Competent framework emphasizesthe heuristic of working backwards x from the goal to the solution (Step 3). Workingbackwards is a powerful heuristic for students who have do not know where to starttheir mathematics solutions when faced with a problem that is slightly different thanthe example problems and solutions they encounter in class. Apparently thefreshman students at Carnegie Mellon University do not use the pattern-matchingnovice strategy as much as our freshmen.Another difference in the two frameworks is how they distinguish physics principlesthat apply in many topics (e.g., kinematics, Newton’s second Law, conservation ofenergy) versus relationships that apply only in specific problem situations (e.g.,constant force, potential energy when the gravitational interaction is near the Earth’ssurface). In Reif’s framework, this difference is not emphasized, whereas in ourCompetent framework the difference is emphasized. Consequently, there are two substepsin the Competent framework (illustrated in Steps 1C and 2C) and only one sub-

Chapter 14: Personalizing a Problem Solving Framework 189step in Reif’s framework (illustrated in Steps 1B and 2B). There are two reasons forthis difference.1. Reif’s framework is limited to one semester of mechanics.2. The prominence of principles is part of our overarching goal of teachingphysics through problem solving. Many students enter our courses believingthat physics is a collection of disconnected concepts and equations. One ofour goals is to help dispel this belief by approaching all topics with the samefundamental physics principles. [Here we go again. Lets see how theconservation of energy applies to electric circuits.]Despite the differences in the two frameworks, the contents of the frameworks areremarkably similar. The two frameworks were developed independently at about thesame time, as was the framework by Alan Van Heuvelen. 1 These problem-solvingframeworks are based on the same problem-solving research, consequently they aresimilar in content, if not in specific heuristics and language.Step . Reconcile Your Framework with Your Textbook Strategy(if necessary)Physics textbooks often include problem-solving strategies that they use in allexample problem solutions. If your textbook has a problem-solving strategy, then yournext step is to figure out how you can use your problem-solving framework inconjunction with the textbook strategy.Compare the Textbook Strategy with Your Framework. Examine the problemsolvingstrategy in your textbook and compare it with your framework. How isstrategy similar to your framework? How is it different?Identify Advantages and Disadvantages of Textbook Strategy. The next step is toidentify the features of the textbook strategy that are advantages and those thatconflict with your problem-solving goalsReconcile. Build on the advantages of the textbook strategy and modify yourframework to at least make use some of the same language as the textbook strategy.An ExampleCompare the Textbook Strategy with Your Framework. One physics textbook usesa Picture, Solve, and Check problem-solving format. xi An examination of thistextbook strategy resulted in the following similarities and differences with FredReif’s research-based framework (Figure 14.7).1. The authors of this textbook do not provide general guidelines for whatstudents should think about or do within each step. In some chapters the authors

190 Part 4: Personalize a Problem Solving Framework and ProblemsFigure 14.8. Example of a specific problem-solving strategy in a textbookApplying Newton’s Second LawPICTURE Make sure you identify all of the forces acting on a particle. Then determine thedirection of the acceleration vector of the particle, if possible. Knowing the direction of theacceleration vector will help you choose the best coordinate axes for solving the problem.SOLVE1. Draw a neat diagram that includes the important features of the problem.2. Isolate the object (particle) of interest, and identify each force that acts on it.3. Draw a free-body diagram showing each of these forces.4. Choose a suitable coordinate system. If the direction of the acceleration vector isknown, choose a coordinate axis parallel to that direction. For objects sliding along asurface, choose one coordinate axis parallel to the surface and the other perpendicularto it.5. Apply Newton’s second law, F ma , usually in component form.6. Solve the resulting equations for the unknowns.CHECK Make sure your results have the correct units and seem plausible. Substitutingextreme values into your symbolic solution is a good way to check your work for errors.include a problem-solving strategy, but each is limited to a particular topicand/or problem constraint. For example, the strategy shown in Figure 14.8 islimited to contact forces on one solid object that can be considered a particle.Other specific strategies found within three chapters were: Applying Newton’sSecond Law to Problems with Two or More Objects; Solving ProblemsInvolving Friction; and Solving Problems Involving Work and Kinetic Energy.In contrast, research-based frameworks emphasize deciding the approach totake, starting with fundamental principles, and adding the problem constraintsthat apply to the problem situation.2. The three major steps in the textbook strategy (Picture, Solve, Check) aresimilar to, but not the same as, the major steps in the framework by Fred Reif:Analyze the Problem, Construct a Solution, and Check and Revise.• The Picture step in the textbook strategies include hints about what to besure to remember in the limited problem constraint. It assumes thatstudents already know what concepts and principles to apply (Reif’s Step 1:Basic Description and Refined Description.• The Solve step in the textbook strategy overlaps Step 1 in Reif’sframework. Moreover, the information in the Solve step does not includeany heuristics on planning a solution and executing the plan.• The Check step in the same as the Check and Revise step in Reif’sframework.

Chapter 14: Personalizing a Problem Solving Framework 191Identify Disadvantages of Textbook Strategy. The next step is to identify thefeatures of the textbook strategy that are contrary to your specific problemsolvinggoals. The major disadvantage is that students cannot generalize thetextbook strategies across topics. xii In fact, the strategies reinforce studentspattern-matching novice strategies.Reconcile. Build on any advantages with the textbook strategy. For example,you could take your framework and divide into the three major of Picture, Solve,and Check (i.e., use the same names as in the textbook). You could introduceyour framework as an expansion or generalization of the specific strategies inthe text.Step . Evaluate Your FrameworkRemember the Zeroth Law of Instruction (If you don’t grade for it, students’ won’t do it.)and the 1st Law of Instruction (Doing something once is not enough.). Introduce yourclass to your problem-solving framework and require that your students use it.Model the entire framework whenever you solve problems for the class. Usecontext-rich problems on tests that require the use of your organized and logicalproblem-solving framework (see Chapter 3 and Chapter 9, pages //-//). Studentsalso need to be able to practice solving context-rich problems before the tests, bothindividually and in cooperative groups (see Chapter 9, pages //-//.Examine samples of students’ problem solutions.1. Focus on solutions in which students follow a logical and correct path up to apoint but cannot correctly solve the problem.2. Determine if there are common places in the framework where a significantnumber of students either make a jump to some incorrect or illogical solutionpath or simply stop. These are candidates for additional sub-steps in yourproblem-solving framework.3. Examine existing research-based frameworks to see if they address this issue.If so, add that part of the framework to your own. If not, invent a sub-stepthat you believe would allow the student to continue following their logicalpath to a correct solution. These sub-steps should not be topic specific. Ifthere is a physics difficulty, address that directly in your class using othertechniques.EndnotesiSee, for example, Maloney, D.P. (1994), Research on problem solving in physics.In D.L. Gabel (Ed.), Handbook of research in science teaching and learning, (pp. 327-354), NY, Macmillan; Leonard, W.J., Dufresne, R.J., & Mestre, J.P. (1996), Using

192 Part 4: Personalize a Problem Solving Framework and Problemsqualitative problem-solving strategies to highlight the role of conceptualknowledge in solving problems, American Journal of Physics, 64(12): 1495-1503: andPretz, J. E., Naples, A. J., & Sternberg, R. J. (2003), Recognizing, defining, andrepresenting problems. In J. E. Davidson & R. J. Sternberg (Eds.), The psychology ofproblem solving (pp. 3-30). Cambridge, UK: Cambridge University Press.iiWe randomly selected 100 student solutions from each of three large sections ofthe calculus-based course, using the proportion of students in each section whoreceived the grades of A, B, C, and D.iii Sabella, M. and Redish, E.F. (2007). Knowledge organization and activation inphysics problem polving, American Journal of Physics, 75, 1017-1029.iv Reif, F. (1995). Understanding basic mechanics: Text, NY: Wiley.vVan Heuvelen, A. (1991), Overview, case study physics, American Journal of Physics,59(10), 898-907.vi Leonard, W.J., Dufresne, R.J., Gerace, W.J. & Mestre, J.P. (2002). Minds•On Physics,NJ: Kendall Hunt.vii Polya, G. (1957). How to solve it: A new aspect of mathematical method. Princeton, NJ:Princeton University Press.viii Heller, K. & Heller, P. (2000). The competent problem solver for introductory physics: Calculus,McGraw-Hill Higher Education.ix Heller, K. & Heller, P. (2000). The competent problem solver for introductory physics: Algebra,McGraw-Hill Higher Education.xThe working backwards heuristic starts with the ultimate goal and then decidingwhat would constitute a reasonable step just prior to reaching that goal. Thenask yourself what the step would be just prior to that, and so on until you reachthe initial conditions of the problem.xi Tipler, P.A., and Mosca, G. (2008). Physics for scientists and engineers: 6th edition, NY:WH Freeman.xii Research over the last three decades indicates that students cannot learncomplex skills, such as reading comprehension and problem solving by practicingspecific sub-skills. It is a case of the complex skill being more than the sum ofits sub-skills. This was the motivation behind the cognitive apprenticeshiptheory of instruction, which emerged after research studies of readingcomprehension and problem solving in mathematics. In cognitive apprenticeshipan expert models (demonstrates) the entire complex skill so students can form aninitial, conceptual idea of the skill. Scaffolding and coaching as students practicethe complex skill, sometimes concentrating on a sub-skill, accompany themodeling. Finally, the scaffolding and coaching is gradually faded. See Chapter 6for additional explanations and references.

193Chapter 15Building Context-rich Problemsfrom Textbook ProblemsIn this chapter: What are the properties of context-rich problems? How you can build your own context-rich problems based on textbook problems. Practice constructing context-rich problems.In Chapter 2 we introduced the idea of real problem solving -- the process ofarriving at a solution when you don’t initially know what to do. Real problemsolving involves making decisions in analyzing the problem situation, decidingwhat fundamental principles to apply, and deciding the information is needed tosolve the problem. Most novice students tend to memorize rather that engage inlogical decision making. They tend to plug numbers into memorized formulas andmanipulate the formulas mathematically until they get an answer. Or they maymemorize patterns of equations to solve different classes of problems (e.g., free fallproblems, inclined plane problems). Traditional end-of-chapter problems do notpromote the development of skills necessary to solve real problems.In Chapter 3 we discussed how context-rich problems help student engage in realproblem solving in order to improve their problem solving skills. In Chapter 8 wedescribed the function of context-rich problems in cooperative group learning. Thischapter suggests a procedure and provides some practice examples for you toconstruct context-rich problems that fit the needs of your students.

194 Part 4: Personalize a Problem-solving Framework and ProblemsReview of the Properties of Context-rich ProblemsEvery context-rich problem has common properties that reinforce the developmentof students’ problem-solving skills. These properties, described in more detail inChapter 3 (pages // - //), are outlined below.Short Story. The problem is a short story, in everyday language, in which themajor character is the student. That is, each problem statement uses thepersonal pronoun “you.”Motivation. The problem statement includes a plausible motivation or reasonfor “you” to calculate something.Real Objects. The objects in the problems are real (or can be imagined) -- theidealization process occurs explicitly.No Diagrams. No pictures or diagrams are given with the problems. Studentsmust visualize the situation by using their own experiences and knowledge ofphysics.Two or More Steps. The problem solution requires more than one step oflogical and mathematical reasoning. There is no single equation that solves theproblem.Fundamental Principles. The problem can be solved by the straightforwardapplication of a fundamental principle of physics. (e.g. Newton’s laws,conservation of energy).In addition the problem difficulty can be adjusted to make the problem suitable forindividual or group work. We have identified twenty-one traits that make problemsmore difficult to solve. These traits are explained in Chapter 16. A few commondifficulty traits are listed below.No explicit target variable. The unknown quantity is not explicitly specified inthe problem statement (e.g., Will this plan (design) work? or Should you fight thistraffic ticket in court?).Excess Data. More information is given in the problem statement than isrequired to solve the problems (e.g., the inclusion of both the static and kineticcoefficients of friction).Unusual Ignore/Neglect Assumption(s). The problem solution requires studentto ignore a small effect (e.g., rotational energy of a pulley), or neglect a small,obvious effect (e.g., mass of an object other than the mass of a string).Two Fundamental Principles. The problem requires more than one fundamentalprinciple for a solution (e.g., kinematics and the conservation of energy).

Chapter 15: Building Context-rich Problems from Textbook Problems 195Figure 15.1. Important problem features and properties of context-richproblems that support these features.Problem Features ThatPromote the Developmentof Problem-solving SkillsFeature 1. It is difficult to usenovice strategies of pluggingnumbers into formulas ormatching a solution pattern toget an answer.Feature 2. It is difficult to solvethe problem without firstanalyzing the problem situation.Feature 3. Physics cues, such as“inclined plane”, “starting fromrest”, or “projectile motion”, areavoided in order to help studentspractice connecting physicsknowledge to other things theyknow.Feature 4. The problemreinforces a logical analysisusing fundamental principles.Properties of Context-rich Problemsthat Support the FeaturesAll Problems• The problem is a short story, in everyday language, inwhich the major character is the student (“you”).• The objects in the problems are real (or can beimagined) -- the idealization process occurs explicitly.• The problem can be solved by the straightforwardapplication of a fundamental principle of physics.More Difficult Problems• More information is given than is needed to solve theproblem.• Problem solution requires student to neglect a small,obvious effect or ignore a small effect (assumptions).All Problems• The problem is a short story, in everyday language, inwhich the major character is the student (“you”).• The problem statement includes a plausiblemotivation or reason for “you” to calculatesomething.• No pictures or diagrams are given with the problems.More Difficult Problems• The unknown quantity is not explicitly specified in theproblem statement (e.g., “Will this plan work?”)• The problem requires more than one fundamentalprinciple for a solution (e.g., conservation of energyand kinematics).All Problems• The objects in the problems are real (or can beimagined) -- the idealization process occurs explicitly.More Difficult Problems• The problem solution requires using the geometry ofthe physical situation to generate a necessarymathematical expression.All Problems• The problem solution requires more than one step oflogical and mathematical reasoning. There is nosingle equation that solves the problem.• The problem can be solved by the straightforwardapplication of a fundamental principle of physics.More Difficult Problems• The problem requires more than one fundamentalprinciple for a solution (e.g., conservation of energyand kinematics).

196 Part 4: Personalize a Problem-solving Framework and ProblemsNecessary Relationships from Diagram: The problem solution requires usingthe geometry of the physical situation to generate a necessary mathematicalexpression.Figure 15.1 shows how some of the common properties and difficulty traits ofcontext-rich problems that are related to the feature of problems that promote thedevelopment of problem solving skills.The first thing people notice about context-rich problems is the use of the personalpronoun “you” and the inclusion of a plausible motivation or reason for “you” tocalculate something. One reason to do this is because it makes the problemsituation easier to visualize and more personal.Using the personal pronoun “you” and providing motivations reinforces the value ofphysics in students’ lives. Most students, especially students who are not physicsmajors, are not intrinsically motivated to study physics and see no value in beingtold to calculate an acceleration or electric field -- “Who cares?” We have foundfour categories of contexts/motivations that reinforce the value of physics instudents’ lives.CuriositySolving context-rich problems should help students realize that the application offundamental principles can satisfy their curiosity about objects and situations theydid not think they could understand. Some motivations in this category include:• You are . . . . (in some everyday situation) andneed/want to figure out . . . .• You are watching . . . . (an everyday situation) andwonder . . . .• You are on vacation and observe/notice . . . . andwonder . . . .• You are watching TV (or reading an article) about .. . and wonder . . .HelpfulnessAt the same time, context-rich problems can help students realize that physics canbe useful both their student lives and the adult life they can envision. Somemotivations in this category include:• Because of your knowledge of physics, your friend asks you to help . . .;• You are helping to design the opening ceremony for the next winterOlympics. One of the choreographers envisions . . .. You have been assignedthe task of determining the minimum speed that the skater must have to . . ..• Because of your interest in the environment and your knowledge of physics,you are a member of a Citizen's Committee (or Concern Group) investigating. . …• You are volunteering with an ecology group investigating the feeding habitsof eagles. During this research, you observe an eagle . . .. You wonder . . ..

Chapter 15: Building Context-rich Problems from Textbook Problems 197Job RelatedMany college students are motivated by the prospect of getting a summer or parttime job, and can envision getting a job when they graduate.• You have a summer job with a company that . . ..Because of your knowledge of physics, your boss asksyou to . . ..• You have been hired as a technical advisor for a TV (ormovie) production to make sure the science is correct.In the script . . . ., but is this correct?• You are a member of a team designing a new device to help trucks go downsteep mountain roads at a safe speed even if their brakes fail. . .. The device is. . .. To help determine these forces, you decide to calculate . . ..Application to unusual or unfamiliar objects or situationsContext-rich problems can also help students realize that a small number of physicsprinciples apply to many situations, both everyday and unfamiliar, and at manydifferent scales (atomic to the solar system).• You have been hired by a college research group that isinvestigating (e.g., cancer prevention, the possibility ofproducing energy from fusion) . . . Your job is todetermine. . . .”• You are reading a magazine article about a satellite inorbit around the Earth that detects X-rays coming fromouter space. The article states that the X-ray signaldetected from one source, Cygnus X-3 . . . You realize that if this is correct,you can determine how much more massive the Cygnus X-3 neutron star isthan our Sun. . ..• You are working for a chemical company with a group trying to produce newpolymers. You have been asked to help determine the structure of part of apolymer chain. . . Your boss wants to know . . . You boss asks you tocalculate . . ..Any context-rich problem will not motivate all students. You should strive formotivating at least half of the students at least half of the time.Figure 15.2. Steps for building a context-rich problem from a textbook problem.Step Decide on the goals of the problem. Choose the fundamental physics principle or principles to be featured. Decide on a difficulty level (e.g., individual or group practice or examproblem, just before or after students have studied principle or concept). Do you want students to confront and resolve a specific misconception?

198 Part 4: Personalize a Problem-solving Framework and Problems Do you want students to practice a specific technique (e.g. calculate the components ofvectors, determine a functional dependence, compare a number with their experience)?Step Construct the problem on the foundation of one that already exists. Find a textbook problem that could satisfy your goals. If the problem does not have one, invent a context (real objects with real motions orinteractions) that seems natural for that problem. Decide on a motivation -- Why would the student want to calculate something in this context? Determine if you need to change the quantity to be calculated or the input quantities to: make the solution involve more one logical step using more than one equation; and correspond more naturally with the context or motivation.Step Write the problem as if it were a short story that connects to your students’ reality. Put the student into the problem. Make sure the context is as gender and ethnic neutral as possible. Use the motivation and context to reinforce the value of physics to the student. Job related (e.g., “You have a summer job . . .”) Curiosity (e.g., “You wonder why . . .”) Helpfulness (e.g., “You are helping your friend . . . ; “) Application to unusual or unfamiliar objects or situations (e.g., molecules instead of cars) Give the student the opportunity to make decisions.Some suggestions for creating a problem with more decisions are (see Chapter 16 for moredifficulty traits): Add extra information that someone in that situation would be likely to have. Leave out necessary but common-knowledge information (e.g. the boiling temperature ofwater). Write the problem so the target quantity is not explicitly stated but must be calculated tomake a decision. Expand the problem so that two different major physics principles are needed. Change to unfamiliar objects or situations (e.g. change cars to electrons). Make the problem as short as possible by eliminating unnecessary words or descriptions.Make sure the problem is not obscured by a fog of unnecessary words or information.Step Check to make sure that: the solution requires more than one logical step and one equation; the solution proceeds directly from fundamental principles and requires no subtle insights orsubtile mathematics.Step Check evaluation possibilitiesMake sure the student has a reasonably straightforward way to check the correctness of theanswer by: Checking the units of the answer. Comparing to similar quantities that the student should know; and/or Taking the function to limits where the student knows the behavior (calculus-based course).

Chapter 15: Building Context-rich Problems from Textbook Problems 199Constructing a Context-rich Problem from a Textbook ProblemYou can always construct a context-rich problem from scratch, but the easiestway is to start from a textbook exercise or problem. Figure 15.2 contains achecklist for five steps to guide the construction of a context-rich problem froma textbook exercise or problem. The steps are explained below, using thetextbook problem shown in Figure 15.3a.Step Decide on the Goals of the Problem. Choose the fundamental physics principle or principles to be featured.Imagine that you want students to practiceusing Newton’s 2nd law as applied touniform circular motion. Decide on the difficulty level (e.g.,individual or group practice or examproblem, just before or after principlesand concepts have been studied)Group problems should be more difficultthan individual problems. In addition, the problem must be easier for usejust after a concept is introduced than for use towards the end of instructionon the concept.Suppose you intend to use a context-rich problem as a group practiceproblem just after you introduced the connection between a constant forcetowards the center of the circle and centripetal acceleration. Since you havenot yet modeled solving problems with more than one force, you need aproblem situation with only one force. To make the problem more difficultfor a group, however, you decide to use a problem involving the vectorcomponents of a force. Do you want students to confront and resolve a specific misconception?Solving context-rich problems, especially in collaborative groups, can helpstudents confront and overcome their alternative physics conceptions, calledmisconceptions for simplicity. In this case however, no specificmisconceptions are targeted. Do you want students to practice a specific technique (e.g. calculate thecomponents of vectors, determine a functional dependence, compare anumber with their experience, solve simultaneous equations)?Imagine that you have noticed that your students need more practicevisualizing a problem situation and calculating vector components.

200 Part 4: Personalize a Problem-solving Framework and ProblemsFigure 15.3. A textbook and context-rich problem of the textbook problem.Figure 15.3a. A textbook problemIf an aircraft is banked during a turn in level flight at constant speed,the force Fa exerted by the air on the aircraft is directed perpendicularto a plane that contains the aircraft's wings and fuselage. Draw a freebodydiagram for such an aircraft. (Hint: Note the similarity to theconical pendulum example in the Chapter 6.) An aircraft traveling at aspeed v = 75 m/s makes a turn at a banking angle of 28 o . What is theradius of curvature of the turn?FamgFigure 15.3b. The context-rich Airplane Problem based on the textbook problemYou are flying to a job interview when the pilot announces that there are airport delays andthe plane will have to circle the airport. The announcement also says that the plane willmaintain a speed of 400 mph at an altitude of 20,000 feet. To pass the time, you decide tofigure out how far you are from the airport.You notice that to circle, the pilot "banks" the plane so that the wings are oriented at about10 from the horizontal. An article in your in-flight magazine explains that an airplane can flybecause the air exerts a force called "lift" on the wings. The lift is always perpendicular to thewing surface. The magazine article gives the weight your type of plane as 100,000 poundsand the length of each wing as 150 feet. It gives no information on the thrust from theengines or the drag on the airframe.Step Construct the Problem on the Foundation of One That AlreadyExists. Find a textbook exercise or problem that could satisfy your goals.Imagine that you found the textbook problem shown in Figure 15.3a. Thisis the beginning of a good problem. It requires the application ofNewton’s 2 nd law using the components of a single force and theconnection of those forces to circular kinematics. If the problem does not have one, invent a context (real objects with realmotions or interactions) that seems natural for that problem.In this case, the textbook problem in Figure 15.3a has a reasonable objectand interaction -- the force of the air (lift) on a banked airplane in uniformcircular motion. Some suggestions for interactions and objects for otherproblems include: Physical work (pushing, pulling, lifting objectsvertically, horizontally, or up and down hills or ramps) Suspending objects, falling objects Sports situations (falling, jumping, running, throwing,etc. while diving, bowling, playing golf, tennis, football,baseball, etc.)

Chapter 15: Building Context-rich Problems from Textbook Problems 201 Situations involving the motion of bicycles, cars, sailboats, trucks, airplanes, windmills, etc. Astronomical situations (motion of satellites, planets) Rotating objects (wheel, doors on hinges, flywheel,sewer pipe rolling down ramp, large spools in a factory,etc.) Circuits in battery operated devices and home circuits(e.g., for lamps, kitchen appliances) More complicated circuits in devices (for applicationof conservation or charge and energy) Charged particles (e.g., pollen in ionization filter, electrostatic scale,special devices) Electric and Magnetic Fields (investigating CRT screen in lab, conductingbar sliding on two parallel conducting rails in safety device, Helmholzcoils in various devices, etc.) Decide on a motivation -- Why would the student wantto calculate something in this context?Make the student a passenger of the airplane. Tocomplete the motivation, answer the following questions:Some example answers are included.• Why is the student in an airplane? Make the student a passenger.• Why does the airplane circle? There is an airport delay and the airplanemust circle the airport before it can land.• Why should students calculate the radius of the circle? The student wondershow far the airplane is from the airport.• Where does the student get the necessary information? The student can estimatethe banking angle. The only information available to a passenger is anannouncement from the pilot and, perhaps, the in-flight magazine. Determine if you need to change the quantity to be calculated or the inputquantities to:• make the solution more one logical step using more than one equation;• correspond more naturally with the context or motivation.In this case, there is no need to change the quantity to be calculated.Step Write the Problem as If It Were a Short Story That Connects toYour Students’ Reality.In writing a story, eliminate the physics words and symbols so that the studentgets practice supplying them. For example, in the context-rich airplane problem(Figure 15.3b), we eliminated the words constant speed and banking angle, and thesymbol Fa. In addition, the units of quantities in a textbook problem may need

202 Part 4: Personalize a Problem-solving Framework and Problemsto be changed to be consistent with the context. This information should bewhat would be naturally available in that context. For example, the velocity ofthe airplane was changed from meters per second to miles per hour. The weightof the airplane is in the common units of pounds, and the distance and lengthare in the common units of feet. Put the student into the problem.Personalize and motivate the problem for the students by making thestudent the major character in the story (use the pronoun “you”). “You”also contributes to making problems gender neutral. Make sure the context is as gender and ethnic neutral as possible.While flying in a commercial airplane is gender and ethnic neutral, not allstudents have this experience. However, most students have experiencewatching movies or TV shows involving flying. Use the motivation and context to reinforce the value of physics to thestudent.Another way to motive students is to name local landmarks in your problem,such as buildings, lakes, parks, or shopping malls. You can also includecharacters in popular TV shows or movies, and use movie names that arehumorous take offs of popular current movies.We also found that successful motivations are different for differentintroductory courses. For example, our algebra-based class consists ofstudents majoring in architecture (40%), environmental science, forestry,veterinary medicine, and health-related fields such as food science andnutrition, nursing and pharmacy. On the average, these students are olderthan students in the calculus-based course for scientist and engineers, andabout one-half of the students are women. They tend to like the curiosityand helpful motivations, and some practical job-related motivations.The freshman in the calculus-based course are mostly engineering majorsand tend to like the curiosity and job-related motivations, with moreapplications with unusual objects and situations. In our course for biologymajors, we constructed many problems with contexts and motivationsrelevant to biology. Give the student the opportunity to make decisions.Eliminate the physics words and symbols so that the student gets practicesupplying them. Eliminate the diagram from the textbook problem and letthe student decide on the best way to picturethe airplane. Eliminate the hint of making afree-body diagram and the hint about theconical pendulum.To make the problem more suitable forcooperative problem solving, add somecharacteristics that require groups to make more

Chapter 15: Building Context-rich Problems from Textbook Problems 203decisions. Notice that the statement of the context-rich airplane problemforces a connection with reality because the group must decide what “howfar you are from the airport” means. The problem also gives moreinformation than necessary to force the group to use physics principlesrather than churn all the numbers that are available.Beware: There is a tendency to make group problems too difficult. A goodgroup problem does not have all of the characteristics that make a problemmore difficult, but usually only a few of these characteristics (see Chapter16). Make the problem as short as possible by eliminating unnecessary wordsor descriptions. Make sure the problem is not obscured by a fog ofunnecessary words or information.This is one of the most difficult characteristics to achieve. The eliminationof a picture or diagram requires a clear and concise description of theproblem situation. This increases the length of context-rich problemscompared to most textbook exercises or problems. The motivation of theproblem also needs to be as short as possible. For example, in the contextrichairplane problem (Figure 15.3b), the motivation and some of thenecessary information is given in the first three sentences.Step Check the Solution to Make Sure That: the solution requires more than one logicalstep and one equation; the solution proceeds directly fromfundamental principles and requires no subtleinsights or special mathematics.The problem can be solved with astraightforward application of Newton’s 2ndlaw in the vertical and horizontal directions.The most difficult part of the problem isdrawing the free-body diagram and relating thebanking angle to the components of the lift force.Step Check Evaluation PossibilitiesMake sure the student has a reasonably straightforward way to check thecorrectness of the answer by: checking for correct units comparing to similar quantities that the student should know; taking the function to limits where the student knows the behavior.

204 Part 4: Personalize a Problem-solving Framework and ProblemsThe function for radius indicates that the radius decreases as the banking angleincreases, and increases as the velocity increases. Both of these relationshipsare consistent with experiences in the circular motion lab.Practice Building Context-rich ProblemsPractice Example 1: Electric Field and Gauss’ LawAs a change of pace, practice constructing a context-rich problem for the secondsemester of a calculus-based introductory physics course -- a more sophisticatedproblem involving electric fields and Gauss’s Law. This example assumes that thestudents have already had one semester of a physics course based on real problemsolving.There are two ways you could use this example. You could simply read through theexample to learn more about the procedure for building a context-rich problemfrom a textbook problem. Or you could read Step (Decide the goals of theproblem) below, accept or modify these goals, then use the guidelines in Figure 15.2to practice constructing your own context-rich problem based on the textbookproblem in Figure 15.4.Step Decide on the Goals of the Problem. Choose the fundamental physics principle or principles to be featured.The major goal is for students to practice using Gauss’ Law for electricfields to calculate something interesting about the behavior of real chargedobjects by using it in conjunction with mechanics. Decide on a difficulty level. This will be a difficult problem, suitable forstudents working in groups. Do you want students to confront and resolve a specific misconception?No known misconceptions will be addressed, but the concept of Gauss’sLaw is very difficult for some students to grasp. Determine if you need to change the quantity to be calculated or the inputquantities. Students practice:• choosing a simple Gaussian surface and determining the charge insidethat surface from a charge density;• using concepts from the first semester of the course, in this caseconservation of energy;• doing appropriate integration. Practice using numbers to get used toappropriate magnitudes for realistic charged objects.

Chapter 15: Building Context-rich Problems from Textbook Problems 205Figure 15.4. A textbook exercise and context-rich problem about electric fieldsFigure 15.4a. A textbook exercise about electric fields and Gauss’s LawAn infinitely long cylinder of radius R carries a uniform (volume) charge density r. Calculate thefield everywhere inside the cylinder.Figure 15.4b. The context-rich Fusion Problem based on the textbook exerciseYou have a job in a research laboratory investigating the possibility of producing power fromfusion. The device being tested confines a hot gas of positively charged ions in a very longcylinder with a radius of 2.0 cm. The charge density in the cylinder is 6.0 x 10 -5 C/m 3 .Positively charged Tritium ions are to be injected into the cylinder perpendicular to its axis ina direction toward its. Your job is to determine the speed that a Tritium ion should havewhen it enters the cylinder, so that it just reaches the axis of the cylinder.Tritium is an isotope of Hydrogen with one proton and two neutrons. You look up the chargeof a proton and mass of the tritium in your trusty Physics textbook and find them to be 1.6 x10 -19 C and 5.0 x 10 -27 Kg.Step Construct the Problem on the Foundation of One That AlreadyExists. Find a textbook problem that could satisfy your goals. The textbookexercise in Figure 15.4a could be modified to meet the goals. If the problem does not have one, invent a context. One natural context isa cylindrical volume containing a charge density could be part of acontainment vessel. Decide on a motivation. A possible motivation is that “you” want to knowthe functional form of the field so that you can inject a charged particle intothe existing ions. “You” might want to inject the charged objects (call itTritium) to investigate fusion as a source of power. Determine if you need to change the quantity to be calculated or the inputquantities. Change the calculated quantity to the speed that a chargedobject must have to penetrate to the center of the cylinder. Specify theradius of the cylinder and the charge density as in the original exercise. Addthe charge and mass of the object to be shot into the cylinder.Step Write the Problem as If It Were a Short Story That Connects toYour Students’ Reality. Put the student into the problem Use the motivation and context toreinforce the value of physics. The context and motivation for the contextrichproblem in Figure 15.4a is job related: You have a job in a researchlaboratory investigating the possibility of producing power from fusion.

206 Part 4: Personalize a Problem-solving Framework and ProblemsThe device being tested . . . . Your job is todetermine the speed that a Tritium ion shouldhave when it enters the cylinder, so that it justreaches the axis of the cylinder. The context isgender and ethic neutral for students in acalculus-based course for scientists and engineers. Give the student the opportunity to makedecisions. The students must decide to useconservation of energy and then decide to useGauss’ Law to find the force on the Tritium at every point in its flight. Thestudent must also decide to assume that the charge density of the existingions is uniform.Step Check the Problem Solution to Make Sure That: The solution requires more than one logical step and one equation. Thesolution requires equations for Gauss’ Law, conservation of energy, and thecharge density. It is not a one-step problem. The solution proceeds directly from fundamental principles and requiresno subtle insights or subtile mathematics. A simple path of the Tritium isspecified so that the application of the physics principles is straightforward.Only a simple integral is required. This is a sophisticated problem but theapplication of Gauss’ Law, conservation of energy, and the mathematics isnot subtle.Step Check Evaluation PossibilitiesMake sure the student has a reasonably straightforward way to check thecorrectness of the answer by: Checking the units of answer. The units of the answer can be determinedas a check. Comparing to similar quantities that the student should know. Themagnitude of the answer can be compared with the speed of light on thefast side. Taking the function to limits where the student knows the behavior. Thealgebraic solution behavior can be checked by increasing the ion chargedensity (increasing repulsion so the injection speed should increase) andincreasing cylinder radius (going a longer distance should require anincreased injection speed).

Chapter 15: Building Context-rich Problems from Textbook Problems 207Practice Example 2.At the other end of the difficulty scale, construct a context-rich problem suitable forkinematics at the beginning of an introductory physics course. This problemassumes that students are just beginning problem solving.Step Decide on the Goals of the Problem. Choose the fundamental physics principle orprinciples to be featured. Practice usingsimple two-dimensional kinematicsemphasizing the independence of orthogonalcomponents of motion. Decide on a difficulty level. This will be aneasy problem suitable for students workingalone. Do you want students to confront a specificmisconception? Several misconceptions willbe addressed.• The velocity does not depend on the massof the object falling.• The time an object takes to fall is independent of its horizontal velocity. Do you want students to practice a specific technique? Students shouldpractice using a coordinate system and choosing its origin. Students practicegetting a solution without using numbers.Step Construct the Problem on the Foundation of One That AlreadyExists. Find a textbook problem that could satisfy your goals. Start with thetextbook problem in Figure 15.5a. If the problem does not have one, invent a context. Use the rifle in thecontext of hunting. Decide on a motivation. The motivation is a crime scene investigation.There is a possible hunting accident. “You” need to determine therelationship between where the bullet hits the ground and its muzzlevelocity to set up special effects for a movie. Determine if you need to change the quantity to be calculated or theinput quantities. Have students practice solving a problem withoutnumbers.

208 Part 4: Personalize a Problem-solving Framework and ProblemsFigure 15.5. A textbook and context-rich problem for two-dimensional motionFigure 15.5a. A textbook problem about two-dimensional motionA rifle is aimed horizontally at a target 100 ft away. The bullet hits the target 0.75 in. belowthe aiming point. (a) What is the bullet’s time of flight? (b) What is its muzzle velocity?Figure 15.5b. The context-rich problem based on the textbook problemYou have a job working on the special effects team for a murder mystery movie. The movieopens with a body discovered in a field during the hunting season. A hunter was seenshooting a rifle horizontally in the same field. The hunter claimed to be shooting at a deer,missed, and saw the bullet kick up dirt from the ground. The detective later finds a bullet inthe ground. In order to satisfy the nitpickers who demand that movies be realistic, thedirector has assigned you to calculate the distance from the hunter that this bullet should hitthe ground as a function of the bullet’s muzzle velocity and the rifle’s height above theground.Step Write the Problem as If It Were a Short Story That Connects toYour Students’ Reality. Use the motivation and context to reinforce thevalue of physics. The context and motivation forthe context-rich problem in Figure 15.5a is againjob related: “You have a summer job working onthe special effects team for a murder mysterymovie . . . . In order to satisfy the nitpickers whodemand that movies be realistic, the director hasassigned you to calculate the distance from thehunter that this bullet should hit the ground as afunction of the bullet’s muzzle velocity and therifle’s height above the ground.” Make sure the context is as gender and ethnic neutral as possible. Thecontext and motivation are gender and ethnic neutral because “you” andthe hunter are not identified by name. Some cultures object to hunting.Similarly, we rarely make “you” the participant in a sport because manystudents may have participated or want to participate in the sport. Give the student the opportunity to make decisions. Eliminate breakingthe problem into parts to give the student freedom to decide on theirapproach. The student must decide that the time for the horizontal motionis the same as the time for the vertical motion. The student must alsodecide on the origin of the coordinate system and to neglect air resistance.

Chapter 15: Building Context-rich Problems from Textbook Problems 209Step Check the Solution to Make Sure That: The solution requires more than one logical step and one equation. Thesolution requires equations for both the horizontal motion of the bullet andits vertical motion. It is not a one-step problem. The solution proceeds directly from fundamental principles and requiresno subtle insights or special mathematics. This is a straightforwardapplication of the definitions of velocity and acceleration as applied to thevertical constant acceleration motion and the horizontal constant velocitymotion. The mathematics is neither subtle nor difficult.Step Check Evaluation Possibilities Checking the units of the answer. The units of the answer can bedetermined as a check. Taking the function to limits where the student knows the behavior. Thebehavior of the algebraic solution can be checked for increasing the muzzlevelocity (increasing muzzle velocity gives greater distance) and increasingthe height (increasing height gives greater distance).

210 Part 4: Personalize a Problem-solving Framework and Problems

211Chapter 16Judging Problem Suitabilityfor Individual or Group WorkIn this chapter: What are the traits of a problems that make them more difficult to solve What are the steps for deciding on the suitability of a problem for different purposes: as anindividual or group problem to use just after or towards the end of students’ study of a topic. Examples of Judging Difficulty LevelFor class time in blocks of 50 minute, instructors of CPS typically create threecontext-rich problems for an exam: one group problem and two individualproblems. The group problem is given in one 50-minute class period. Twoindividual problems, often accompanied by some multiple choice conceptualquestions, are given in another 50 minute class period. [See Chapter 9, page //.]The group problem should be more difficult to solvethan either of the two individual problems. The groupexam problem solving session serves as an intense“study group” for the individual exam. One of theindividual problems is usually relatively easy to solvewhile the other is of medium difficulty. In additioninstructors create one group practice problem eachweek (except exam weeks). Because these problemsoccur when material is first being introduced, theyshould be less difficult than an exam problem.As you can probably imagine, the consistent design of context-rich problems withthe appropriate difficulty level requires some attention. You may find it helpful touse the judging criteria described in this section. The criteria involve identifying andcounting the difficulty traits of a problem, then using a set of “rules of thumb” andyour own specialized knowledge of your class to judge whether the problem’sdifficulty level is appropriate for its intended use.Since difficult mathematics is best practiced by individuals, the increased difficultyfor group problems should be primarily conceptual, not mathematical. Anexception might be if your students have a conceptual barrier to carrying out themathematics. For example, at the beginning of an introductory physics course manystudents have not developed an organized technique for doing problems involvingmany steps of algebra. Such a problem would make an appropriate group practiceproblem at that stage of the course. Generally, problems that involve more

212 Part 4: Personalize a Problem-Solving Framwork and Problemsmathematics than physics or problems that require the use of a shortcut or “trick”do not make good group problems. The best group problems involve thestraightforward application of the fundamental principles (e.g., the definition ofvelocity and acceleration, the independence of motion in the vertical and horizontaldirections) rather than the repeated use of derived formulas. Beware -- it istempting to make group problems too complex and difficult for groups to solve.The first section of this chapter describes 21 problem difficulty traits and theresearch base supporting these traits. The second section describes the four criteriafor judging the suitability of a problem for its intended use: individual practice,individual exam, group practice, or group exam. Finally, the third section providesseveral examples of judging the suitability of problems for their intended use.Problem Difficulty TraitsIn our research we have identified twenty-one traits of a problem that affectstudents’ ability to solve it. i Each of these traits causes difficulty for anintroductory level student because it represents a qualitative difference between anexpert and novice approach to a problem solution. Moving a student toward moreexpert-like problem solving requires them to perform at these difficulty levels.Some of these traits can be found in any problem, while others are more likely toturn up in specific topic areas. A good group problem should have between 2 to 5of these difficulty traits, depending on the purpose of the problem and thebackground of the students.The difficulty traits have been classified into three major categories, each with twoor three subcategories. While these categories are not mutually exclusive, they arehelpful in deciding the total number of difficulty traits for a problem. The threecategories are Approach to the Problem, Analysis of the Problem, and MathematicalSolution. The traits in each category are explained in more detail below.Approach to the ProblemThe traits in the Approach category affect how a student decides which concepts,principles and laws apply to a problem. In traditional problems this is often given tothe students either by a direct statement, such as “the carts have an inelasticcollision” or merely by placing the problem at the end of the chapter under asubheading such as “Inelastic Collisions.” Without such cues, the following 7problem traits can make it more difficult for students to decide how to approach aproblem.1. Problem statement lacks standard cues. Novice problem solvers oftendecide on an approach from concrete “cues” in a problem statement. The twodifficulty traits in this subcategory thwart this tendency.A. No explicit target variable. The target quantity is not explicitlystated. Problems with this difficulty trait typically include statements

Chapter 16:Judging Problem Suitability for Individual and Group Work 213such as: Will this plan (design) work? or Should youfight this traffic ticket in court? (see traffic-ticketproblem, Chapter 8, page //). Novice problemsolvers often use the explicit statement of thedesired quantity (e.g., find a) as a cue to theconcepts and principles they should apply to theproblem. ii Problems without this type of cue aremore difficult for students to solve. iiiB. Unfamiliar context. The context of the problem is very unfamiliar tothe students. If the students have no experience with the objects in acontext, such as molecules or galaxies, they have difficulty creating themental connections from the problem statement to their understanding.This translation is critical for any successful problem solution. iv TheFusion problem (Chapter 15, page //) is an example of a problem that ismore difficult to solve because students’ lack familiarity with the objectsin that context.2. Solution requires multiple connections among principles. Asnovices in physics, most students are not initially adept at connectingrecently learned fundamental principles to other principles they know. Thisis especially true if the principles are several steps removed from their directexperience. The next three difficulty traits are examples of how problemstatements can force students into become fluent with principles.A. Very abstract concept. A central concept required to solve theproblem is an abstraction of another abstract concept. Most physicsconcepts are abstract (e.g., forces, energy, field), while college freshmentend to be concrete thinkers. v Some concepts, which we are calling veryabstract, are themselves an abstraction of an abstract concept. Forexample, electric potential is an abstraction of the abstract concept ofelectric potential energy. Most very abstract concepts, such as magnetic fluxand Gauss’s law, are encountered in electromagnetism. The fusionproblem, Chapter 15, page //) has this difficulty trait.B. Choice of useful principles. The problem has more than onepossible set of useful fundamental principles that could be applied toreach a correct solution. For example, consider a problem with a boxsliding down a ramp. Typically either Newton’s Laws or theconservation of energy will lead to a solution, but the act of decidingwhich to use in a particular problem is difficult for students. Novicestudents tend to try to memorize solution patterns for problems basedon the objects or actions in a problem. For example they try to solve all“object-sliding-down-an-inclined-plane” problems with the samesolution pattern, rather than decide which fundamental principles wouldbe most useful to solve that particular problem. vi Problems that requirestudents to make these decisions are more difficult for students. Forexample, traffic-ticket problem, Chapter 8, page //, has this difficulty

214 Part 4: Personalize a Problem-Solving Framwork and Problemstrait if given on a final exam. Students must decide whether to useforces and kinematics or conservation of energyC. Two Fundamental Principles. The solution requires students to usetwo or more fundamental physical principles. Examples include pairingssuch as Newton’s Laws and kinematics (see traffic-ticket problem,Chapter 8, page //), conservation of energy and conservation ofmomentum, conservation of energy and kinematics, linear kinematicsand torque, or Gauss’s Law and conservation of energy (see fusionProblem, Chapter 15, page //). Combining what the students learnedseveral weeks ago with a current principle is difficult for the beginningstudent, who perceives physics as a set of incoherent topics. vii3. Unfamiliar Application of Concepts and Principles. Studentstypically begin learning new concepts or principles by practicing in situationsthat require only a simple, straightforward application. For example,students learn Coulomb’s Law by solving problems that require thedetermination of the total force on a charge located at known distancesfrom other charges. The two difficulty traits in this subcategory requirestudents to generalize their problem-solving knowledge to situations beyondthose already practiced.A. Atypical situation. The setting, constraints,or complexity is unusual compared topractice problems and examples. That is, theproblem combines objects or interactions in amanner unfamiliar to the student. The fusionproblem (Chapter 15, page //) is an exampleof an atypical situation. This trait challengesthe students’ novice pattern-matching problem-solving technique. ,viiiB. Unusual target quantity. That is, the problem requires students tosolve for a quantity that is usually supplied in their homework problemsor examples. For example, when students first encounter the concept ofwork, the external force is usually supplied and students are asked tocalculate the work. A problem that requires students to determine theexternal force would have this difficulty trait. This trait also challengesthe students’ novice pattern-matching, problem-solving strategy.Analysis of the ProblemThe difficulty traits in the Analysis category tax the novice plug-and-chug andpattern-matching problem solving strategy plunging into equations without takingsufficient time and care to analyze the problem. Problem analysis is the translationof the written problem statement into a complete physics description of the

Chapter 16:Judging Problem Suitability for Individual and Group Work 215problem. It includes a determination of which physics concepts apply to whichobjects or time intervals, specification of coordinate axes, physics diagrams (e.g., avector momentum diagram), assigning symbols to important quantities (includingsubscripts), and the determination of special conditions, constraints, and boundaryconditions. The next 9 traits are all examples of how problems that require a carefuland complete qualitative analysis are more difficult for students to solve.4. Excess or Missing Information. Typical practice problems tend to giveexactly the information necessary. Students believe that manipulating thesevalues will solve the problem. Excess or missing information in a problemthwarts this novice strategy and requires students to analyze the problemsituation, using physics concepts, to decide how to proceed.A. Excess data. The problem statement includes more data than isneeded to solve the problem. For example, the inclusion of both thestatic and kinetic coefficients of friction in the traffic-ticket problem(Chapter 8, page //) require students to decide which frictional force isapplicable to the situation. The airplane problem (Chapter 15, page //)also has excess information (length of airplane wings). Students oftenshow the symptom of novice problem solving by forcing all the giveninformation into the solution.B. Missing Numbers. The problem requires students to either use acommon number, such as the boiling temperature of water, or toestimate a number, such as the height of a person. This case is moredifficult than having too much information because the student mustfirst decide that data is missing and then generate that data byconnecting physics to the rest of their knowledge base.C. Unusual Ignore/Neglect Assumption(s). Theproblem requires students to ignore or neglect asmall effect to solve the problem. All problemswith a context require students to use theircommon sense knowledge of how the worldworks (e.g., boats move through water and notthrough the air.). If assumptions, such as frictionless surfaces ormassless strings, are always made for students in every example orpractice problem, deciding to make this type of simplifying assumptionin a new context will be difficult. For example, most students quicklylearn to ignore air resistance so that assumption does not usually add tothe difficulty level of a problem. On the other hand, ignoring thefrictional force on an ice hockey puck would. There are two classes ofunusual simplifying assumptions: neglect and ignore. The first class isinstances where the students must neglect a quantity that obviously, tothem, makes no difference (e.g. neglecting the mass of a flea whencompared to the mass of a dog). The next class of assumptions involvesignoring small but noticeable effects that cannot be easily expressedmathematically (e.g. the modified Atwood’s machine problem involving

216 Part 4: Personalize a Problem-Solving Framwork and Problemsa string moving downward (see Chapter 5, page //). Although thestring changes the downward force on the system, students must assumea constant downward force to solve the problem.5. Seemingly missing information. The problem requires students togenerate a mathematical expression from the specific situation of theproblem. There are three traits in this subcategory.A. Verbal math statement. The problem includes a verbal statementthat can be expressed as an equation. For example, if the problem states“A is proportional to B,” then the students must translate the writtenstatement into a mathematical expression anddecide how to use it. Examples of verbalmath statements are: “the counter-weight isalways twice the mass of the package on theramp” and the statement in the illustration atright. The fusion problem (Chapter 15, page//) has this trait because the statement,“Tritium is an isotope of hydrogen consistingof one proton and two neutrons” must betranslated into information about charge andmass.B. Special Conditions or Constraints. The problem requires studentsto generate a mathematical expression from the special conditions orconstraints of the problem. An example is the generation of therelationship a 1 = a 2 for two masses connected by a taut string.C. Necessary Relationships from Diagram: The problem solutionrequires using the geometry of the physical situation to generate anecessary mathematical expression. This characteristic adds difficulty toa problem because it emphasizes the necessity of a diagram, a skill manynovice problem-solvers lack. ix6. Additional Complexity. The more “pieces” students have to keep trackof, the more difficult the problem.A. More than two subparts. Some problems require studentsdecompose the problem into more than two sub-parts. More than twosub-parts can arise because there are more than two interacting objectsor more than two important time intervals. Changing systems ofinterest can be hard for students. In addition, novice problem-solversoften lose sight of the problem goal through numerous subparts.Examples of this trait include such classic problems as the ballisticpendulum (which requires using conservation of energy both before andafter the impact, but using conservation of momentum during theimpact), and the massive-pulley Atwood machine (which requiresanalyzing the torques on the pulley and the forces on both suspendedweights).

Chapter 16:Judging Problem Suitability for Individual and Group Work 217B. More than 4 terms per equation. Theproblem involves five or more non-zeroterms in an equation. After a requiredprinciple has been identified by a student(e.g., Newton’s 2nd Law), the principlemust be translated into an equation.Problems that have five or more terms inthat equation push the limits of student’sshort-term memory. x Students musthave a procedural knowledge base that includes organizing principlesand logical connections to solve this type of problem. Typical examplesinclude problems in which 5 or more forces are acting on a single objectalong one axis, and problems in which there are 5 or more energy termsin the conservation of energy equation. Problem statements with thistrait require special care in specifying quantity names and determiningthe sign for each term.C. Vector components. The problem requires students to apply thesame principle (e.g., forces or conservation of momentum) inorthogonal directions. This requires the definition of a coordinatesystem, the decomposition of the physics quantities, and the carefullabeling of those quantities. For example, deciding that it is necessary todetermine electric field vector components is one of the stumblingblocks in integrating the field of continuous charge distributions.Mathematical SolutionMathematical difficulty is last category of traits. Some of these are included inthe last five traits.7. Algebraic solution. A strictly algebraic solution is challenging for manynovice problem-solvers. There are three problem types that can requirealgebraic solutions.A. No numbers. The problem statement does not use any numbers.Expert problem-solvers routinely solve problems without substituting innumbers until the very end. Beginning students, however, do not. Manystudents use numbers as placeholders to help them remember whichquantities are known and which are unknown. xi The mystery movieproblem (Chapter 15, page //) has this difficulty trait.

218 Part 4: Personalize a Problem-Solving Framwork and ProblemsFigure 16.1. The context-rich Airplane ProblemYou are flying to a job interview when the pilot announces that there are airport delays andthe plane will have to circle the airport. The announcement also says that the plane willmaintain a speed of 400 mph at an altitude of 20,000 feet. To pass the time, you decide tofigure out how far you are from the airport.You notice that to circle, the pilot "banks" the plane so that the wings are oriented at about10 to the horizontal. An article in your in-flight magazine explains that an airplane can flybecause the air exerts a force called "lift" on the wings. The lift is always perpendicular to thewing surface. The magazine article gives the weight your type of plane as 100 thousandpounds and the length of each wing as 150 feet. It gives no information on the thrust fromthe engines or the drag on the airframe.B. Unknown cancels. This category includes problems in which an unknownquantity, such as a mass, ultimately factors out of the final solution.Students must not only decide how to solve the problem without all theinformation they expect, but also define symbols for quantities they neitherknow nor can determine.C. Simultaneous equations. This categoryincludes problems in which each unknownquantity cannot be independently determined,as shown in the equations at right. A typicalcircuit-analysis problem best illustrates thistrait. Simultaneous equations are difficult forstudents because of their tendency to solveeach equation for a definite numerical answerbefore moving to another equation. Solving simultaneous equationsalso requires a logical, organized strategy that most beginning studentsdo not initially possess.8. Unfamiliar Mathematics. The problems in this subcategory require studentsto use mathematics that is new and still unfamiliar.A. Calculus or vector algebra. The problem requires calculus, or vectorcross products for a correct solution. Most students are still learning theseskills in their math courses and resist using them in the unfamiliar situationof a physics problem. This is Difficulty Trait 3 applied to mathematics.B. Lengthy or detailed algebra. A successful solution to the problem isnot possible without working through a series of algebraic calculations.While these calculations may not be difficult, they require a carefullabeling of quantities and an organized execution.

Chapter 16:Judging Problem Suitability for Individual and Group Work 219Decision Strategy for Judging the Suitability of Problems for TheirIntended UseImagine that you have just introduced the mechanics of circular motion -- therelationship between the sum of the forces towards the center of a circle and thecentripetal acceleration. Using the criteria for designing context-rich problems, youmay have created the airplane problem (Chapter 15, page //) to use as the firstpractice group problem on this topic. The airplane problem is also shown in Figure16.1. You designed the problem to help students confront and resolve theirconceptual difficulty of associating centripetal acceleration with a single force in thedirection of that acceleration. You also wanted to give students further practice inusing Newton’s 2ndLaw.Figure 16.2. Steps for judging a problem's suitability for an intended use. Read the problem statement. Draw the diagrams and determine the equations needed. Reject if the problem:• can be solved in one step;• involves long, tedious mathematics, but little physics; or• can only be solved easily by using a “trick” or shortcut. Identify and count the difficulty traits of the problem.Problem Approach Analysis of Problem Mathematical Solution1. Cues Lacking A. No target quantity B. Unfamiliar context2. Multiple Connections A. Choice of principle B. Two principles C. Very abstractconcept3. Non-StandardApplication A. Atypical situation B. Unusual targetquantity4. Excess or MissingInformation A. Excess data B. Missing numbers C. Unusual Assumption(ignore/neglect)5. Seemingly MissingInformation A. Verbal Math B. Special constraint C. Relationship fromdiagram6. Additional Complexity A. >2 subparts B. ≥ 5 terms/equation C. Vector Components7. Algebraic solution A. No numbers B. Unknown cancels C. Simultaneousequations.8. Unfamiliar Math A. Calc/vector algebra B. Detailed algebraTOTAL =traits

220 Part 4: Personalize a Problem-Solving Framwork and Problems Use the rules-of-thumb below to decide the suitability of the problem for its intended use.Type of Problem Timing -- Students have DifficultyGroup Practice Problems should beshorter and mathematically less complexGroup Exam Problems can be longer withmore complex math.Individual Problems:EasyMedium-difficultDifficultJust been introduced to concept(s).Just finished study of concept(s).Just been introduced to concept(s).Just finished study of concept(s).Just been introduced to concept(s).Just finished study of concept(s).Just been introduced to concept(s).Just finished study of concept(s)Just been introduced to concept(s).Just finished study of concept(s).2 - 33 - 43 - 44 - 50 - 11 - 21 - 22 - 32 - 33 - 4There are four steps involved in determining whether a problem is suitable to use asan individual or group practice problem at different stages of instruction on a topic.These steps are outlined in Figure 16.2 and described in more detail below.Step If you have not done so, complete at least a partial solution of theproblem -- draw diagrams and determine the equations needed.Step Decide if the problem should be rejected. There are three reasons toreject a problem. First, problems that can be solved with only one equation will notdiscourage students from using their novice plug-and-chug or pattern-matchingtechniques. Second, problems that involve more mathematics than physics makepoor problems for group discussion. Finally problems that require a special shortcutor “trick,” while intriguing, do not emphasize the use of fundamental principles.The Airplane Problem passes the three rejection tests.Step Identify and count the number of problem difficulty traits. This can beaccomplished in three stages: (1) difficulty traits associated with deciding on aproblem approach, (2) traits associated with the analysis of the problem, and (3)traits associated with the mathematical solution. Problem Approach. Determine if the problem has traits that make it difficultfor students to decide on how to start. First compare the airplane problem toyour textbook’s problems for the topic and to problems that you have solvedin class. Analysis of the Problem. Determine if the problem has traits that make itdifficult for students to solve without a careful and complete analysis of thesituation. Mathematical Solution. Determine if the problem has traits that make itdifficult for students to reach a mathematical solution.

Chapter 16:Judging Problem Suitability for Individual and Group Work 221A detailed analysis of the difficulty traits for the Airplane Problem is given in Figure16.3. The airplane problem has a total difficulty rating of 3 - 4 traits.Step The last step is to decide if a difficulty rating of 3 - 4 is appropriate for itsintended use, in this case as a group practice problem. There are three factors toconsider: have about 35 minutes to solve a group practice problem, so it should beless difficult than a Group Exam Problem, where students have about 50 minutes. How much time do the students have to solve the problem? Studentstypically have 20 minutes to solve an individual exam problem, 35 minutes tosolve a group practice problem, and 50 minutes to solve a group examFigure 16.3a. Difficulty traits of the Airplane Problem related to the problem approach.1. Cues Lacking. This problem does not have anexplicit target quantity. Although the problemdoes specify a distance, there are two choicesfor distance -- the radius of the horizontal circleor the actual distance from the plane to theground. The context includes familiar objects(airplane).2. Multiple Connections. The only fundamentalprinciple needed to solve the problem isdynamics (Newton’s Laws).3. Non-standard Application. The situation issimilar to problems found in many textbooks.The problem does not have an unusual targetquantity.Problem Approach1. Cues Lacking A. No target variable B. Unfamiliar context2. Multiple Connections A. Choice of principle B. Two principles C. Very abstract concept3. Non-Standard Application A. Atypical situation B. Unusual targetFigure 16.3b. Difficulty traits of the Airplane Problem related to the analysis of the problem.Analysis of Problem4. Excess or Missing Information. The airplaneproblem statement does include excess data --the length of the plane wings. There is nomissing data, nor is there any unusualignore/neglect assumption that must be made.5. Seemingly missing information. The problemdoes not contain a verbal math statement, andthere is no special constraint. If studentsdecide to solve for the distance from the planeto the ground, then they do need a geometricrelationship from the diagram.6. Additional Complexity. There is one additionalcomplexity trait -- students must use vectorcomponents to solve the problem.4. Excess or MissingInformation A. Excess data B. Missing numbers C. Unusual Assumption(ignore/neglect)5. Seemingly MissingInformation A. Verbal Math B. Special constraintC. Relationship fromdiagram6. Additional Complexity A. >2 subparts B. ≥ 5 terms/equation C. Vector Components

222 Part 4: Personalize a Problem-Solving Framwork and ProblemsFigure 16.3b. Difficulty traits of the Airplane Problem related to the mathematical solutionMathematical Solution7. Algebraic Solution. The airplane problem doesnot require an algebraic solution.8. Unfamiliar Math. The problem does not requireuse of unfamiliar mathematics.7. Algebraic Solution A. No numbers B. Unknown cancels C. Simultaneousequations.8. Unfamiliar Math A. Calc/vector algebra B. Detailed algebraproblem. The problem difficulty should by easy-to medium for theindividual problems, more difficult for the group practice problems, and themost difficult for the group exam problems. Is this a new topic? Problems featuring a topic that has just been introducedshould have fewer difficulty traits than a problem featuring a topic aboutwhich the students have seen example problem solutions. What is the students’ familiarity with both a logical, organized problemsolvingstrategy and with context-rich problems? Problems used early in thecourse should have fewer difficulty traits than those later in the course.Taking these three factors into account, some “rules of thumb” can be used todetermine if your problem is about the right level of difficulty for its intended use.These rules of thumb are shown in Figure 16.2 for this step.If the airplane problem (difficulty rating of 3-4 traits), is intended for use as a grouppractice problem before the students much experience solving uniform circularmotion problems, it would be challenging for most groups to make significantprogress toward a solution in 35 minutes. This problem could be modified byremoving one difficulty trait to make it more appropriate, or could be used as agroup practice problem several days later in the course.There is considerable overlap in the rules of thumb. Many problems can beappropriate for several uses. For example, the Airplane Problem could be used as amedium-difficult individual problem on an exam. As with any procedure, yourjudgment of the appropriateness of a difficulty level should be modified by yourknowledge of your students.Practice Judging Context-rich ProblemsJudging the difficulty level of a problem depends on the students and theinstructor’s view of the course. Even though instructors do not always agree on thespecific difficulty traits of a problem, there is reasonable agreement on the totalnumber of these traits. This is because these criteria are not mutually exclusive. Forexample, it is unusual to find verbal-mathematics statements in problems. In that

Chapter 16:Judging Problem Suitability for Individual and Group Work 223case, some instructors check the Atypical Situation trait, and some check the VerbalMath trait. The only important consideration is not to count the same problemdifficulty twice!Practice Example 1:The Skateboard Problem.Suppose you have just finished teaching about using theconservation of momentum and conservation of energy incollisions. You design the skateboard problem, shown inFigure 16.4, for a group exam problem. The problem isdesigned to test students’ ability to recognize when bothconservation of energy and conservation of momentum areneeded. You may want to determine the suitability of the Skateboard problem for agroup exam problem by following the steps in Figure 16.2 before reading theanalysis below.

224 Part 4: Personalize a Problem-Solving Framwork and ProblemsFigure 16.4. The Skateboard ProblemYou are helping a friend prepare for a skateboard exhibition. The idea is for your friend totake a running start and then jump onto a heavy-duty 15-lb stationary skateboard. Yourfriend, on the skateboard, will glide in a straight line along a short, level section of track, thenup a sloped concrete wall. The goal is to reach a height of at least 6 feet above the startingpoint before rolling back down the slope. The fastest your friend can run and safely jump onthe skateboard is 20 feet/second. Can this plan work? Your friend weighs 125 lbs.The Suitability of the Skateboard ProblemStep Read the problem statement. Draw the diagrams and determine theequations needed. [Note: One solution to this problem is shown in Chapter 11,pages // to //.]Step Decide if the problem should be rejected.The skateboard problem cannot be solved in one step, does not involve long,tedious mathematics, and requires only the straightforward application of theconservation of energy and momentum. It should not be rejected.Step Identify and count the difficulty traits of the problem.The analysis outlined in Figure 16.5 indicates that the skateboard problem has 3-4 difficulty traits: no specific target variable, two principles needed to solve theproblem, unusual assumption required, and possibly the choice of whichprinciples to use.Step Use the rules-of-thumb to decide the suitability of the problem forits intended use.The difficulty rating of 3-4 for the problem indicates it may be too easy for a 50-minute group exam problem for most students. The skateboard problem isprobably more suitable for use as a group practice problem or, perhaps, adifficult final exam problem.Practice Example 2: Gravitational vs Electric ForcesThis practice example problem, shown in Figure 16.5, is from the second semesterof an introductory course. Analyze the problem to determine the number ofFigure 16.5a. Difficulty traits of the Skateboard Problem related to the problem approach.1. Cues Lacking. This problem does not have anexplicit target quantity (Can this plan work?). Thecontext includes familiar objects (people,Problem Approach1. Cues Lacking

Chapter 16:Judging Problem Suitability for Individual and Group Work 225skateboard).2. Multiple Connections. Two fundamentalprinciples are needed to solve this problem --conservation of energy and conservation ofmomentum. Students might spend some timedeciding whether to use the conservation ofenergy or kinematics to relate the velocity of thefriend and skateboard after the inelastic collisionwith the height above the starting point.However, at this point in the course, it should notbe a big issue. A. No target quantity B. Unfamiliar context2. Multiple ConnectionsA. Choice of principle B. Two principles C. Very abstractconcept3. Non-Standard Application A. Atypical situation B. Unusual target3. Non-standard Application. The situation is not atypical of energy and momentum problems,nor is the target quantity unusual. Students can solve for the height to determine if it is lessthan 6 feet, or the initial velocity needed to reach a height of 6 feet and compare it to 20feet/sec.Figure 16.5b. Difficulty traits of the Skateboard Problem related to the problem analysis.4. Excess or Missing Information. The skateboardproblem does not include excess data, nor areany numbers missing. There is, however, anunusual ignore-or-neglect assumption (besidesthe usual assumption of ignoring friction).Students must ignore the vertical component ofthe runner’s momentum during the inelasticcollision with the skateboard. (That is, there is asmall transfer of momentum out of the system.)5. Seemingly missing information. There is noseemingly missing information in this problem.6. Additional Complexity. There is no additionalcomplexity for the skateboard problem. Thereare only 2 subparts (conservation of momentumduring inelastic collision and conservation ofenergy after the collision), 2 terms/equation, andno vectors components.Analysis of Problem4. Excess or MissingInformation A. Excess data B. Missing numbers C. Unusual Assumptions(ignore/neglect)5. Seemingly MissingInformation A. Verbal Math B. Special constraint C. Relationships fromdiagram6. Additional Complexity A. >2 subparts B. ≥ 5 terms/equation C. Vector ComponentsFigure 16.5c. Difficulty traits of the Skateboard Problem related to the mathematical solution.7. Algebraic Solution. The problem does notrequire a strictly algebraic solution -- a number isexpected. There are no unknowns that canceland simultaneous equations are not required fora solution8. Targets unfamiliar math. The math of theproblem is simple and straightforward.Mathematical Solution7. Algebraic Solution A. No numbers B. Unknown cancels C. Simultaneousequations.8. Targets Unfamiliar Math A. Calc/vector algebra B. Detailed algebraFigure 16.6. Gravitational versus Electric Forces ProblemYou and a friend are reading a newspaper article about nuclear fusion energy generation instars. The article describes the helium nucleus, made up of two protons and two neutrons, asvery stable so it doesn’t decay. You immediately realize that you don’t understand why the

226 Part 4: Personalize a Problem-Solving Framwork and Problemshelium nucleus is stable. You know that the proton has the same charge as the electronexcept that the proton charge is positive. Neutrons you know are neutral. Why, you ask yourfriend, don’t the protons simply repel each other causing the helium nucleus to fly apart?Your friend tells you that the gravitational force keeps the nucleus together.Your friend’s model is that the two neutrons sit in the center of the nucleus andgravitationally attract the two protons. Since the protons have the same charge, they arealways as far apart as possible with the neutrons directly between them. It sounds good butyou are not sure you believe it. To check, you decide to determine if the mass the neutronmust have for this model of the helium nucleus to work.Looking in your physics book, you find that the mass of a neutron is about the same as themass of a proton and that the diameter of a helium nucleus is 3.0 x 10 - 13 cm.** Constants, such as the charge of the electron, the universal gravitational constant and the electricconstant, are typically given on an information sheet, but are not included here.difficulty traits. Then determine the possible uses of this problem in an introductorycourse.This problem has a difficulty rating of 4 - 5: an unfamiliar context, requires twoprinciples, an unusual target quantity, and either verbal math or relationshipfrom a diagram. The difficulty is primarily in the visualization of the problem –determining that the distance between the two neutrons is one-half the diameterof the nucleus. It is also possible that the students have not used thegravitational force lately, which makes the problem more difficult.This problem is most suitable for individual practice or a group exam problem,If this were a group problem, you could expect groups to spend most of theirtime discussing the problem situation and the physics to apply. Once that isresolved, the mathematical solution follows easily.EndnotesiFoster T. (2000). The development of students' problem-solving skills frominstruction emphasizing qualitative problem-solving, Ph.D. Thesis: University ofMinnesota.ii Chi M. T. H., Feltovich P. J., & Glaser R. (1981). Categorizations and representationof physics problems by experts and novices, Cognitive Science, 5, 121-152.iii Vollmeyer R., Burns B. D., & Holyoak K. J. (1996). The impact of goal-specificity onstrategy use and the acquisition of problem structure, Cognitive Science, 20, 75-100.iv Larkin J., McDermott J., Simon D.P., & Simon H.A. (1980). Expert and noviceperformance in solving physics problems, Science, 208, 1335-1342.

Chapter 16:Judging Problem Suitability for Individual and Group Work 227v King P. (1986. Formal reasoning in adults: A review and critique in R. Mines & K.Kitchener (Eds), Adult Cognitive Development, Praeger, New York.vi Ref. 1.vii Hammer, D. (1994). Epistemological beliefs in introductory physics, Cognition andInstruction, 12, 151-183.viii Chi M. T. H., Bassok M., Lewis M.W., Reimann P., & Glaser R. (1989). Selfexplanations:How students study and use examples in learning to solve problems,Cognitive Science, 13, 145-182.ix Larkin J. H., & Reif F. (1979). Understanding and teaching problem solving inphysics, European Journal of Science Education, 1 (2), 191-203.x Miller G. A. (1957). The magical number seven, plus or minus two: some limits onour capacity for processing information, Psychology Review, 63, 81-97.xi Wenger R. H. (1987). Cognitive science and algebra learning, in A. H. Schoenfeld(Ed), Cognitive Science and Mathematics Education, Erlbaum, Hillsdale, NJ, pp. 217-252.

228 Part 4: Personalize a Problem-Solving Framwork and Problems

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235Appendix ADepartment Survey1. In your opinion, what is the primary reason your department requires students to take this physics course?2. How many quarters of physics do you think should be required for your students?0 1 2 3 4 5 63. Many different goals could be addressed through this course. Would you please rate each of the followingpossible goals in relation to its importance for your students on a scale of 1 to 5?1 = unimportant 2 = slightly 3 = somewhat 4 = important 5 = veryimportant important importantKnow the basic principles behind all physics (e.g. forces, conservation of energy, ...) 1 2 3 4 5Know the range of applicability of the principles of physics (e.g. conservation ofenergy applied to fluid flow, heat transfer, plasmas, ...)Be familiar with a wide range of physics topics (e.g. specific heat, AC circuits,rotational motion, geometrical optics,...)Solve problems using general quantitative problem solving skills within the context ofphysicsSolve problems using general qualitative logical reasoning within the context ofphysics1 2 3 4 51 2 3 4 51 2 3 4 51 2 3 4 5Formulate and carry out experiments 1 2 3 4 5Analyze data from physical measurements 1 2 3 4 5Use modern measurement tools for physical measurements (e.g.. oscilloscopes,computer data acquisition, timing techniques,...)1 2 3 4 5Program computers to solve problems within the context of physics. 1 2 3 4 5Overcome misconceptions about the behavior of the physical world 1 2 3 4 5Understand and appreciate 'modern physics' (e.g. solid state, quantum optics,cosmology, quantum mechanics, nuclei, particles,...)Understand and appreciate the historical development and intellectual organization ofphysics.1 2 3 4 51 2 3 4 5Express, verbally and in writing, logical, qualitative thought in the context of physics. 1 2 3 4 5Use with confidence the physics topics covered. 1 2 3 4 5Apply the physics topics covered to new situations not explicitly taught by the course. 1 2 3 4 5Other goal. Please specify here 1 2 3 4 5 Please place a star (*) next to the TWO goals listed above that you consider to be the MOSTIMPORTANT for your students.4. In three quarters it is impossible to cover every topic in physics, so some topics need to be left out. Thepurpose of this question is to inform us of your priorities of the topics we might cover in the course. Below

236 Appendix Aare the chapter headings from a typical textbook at this level. Please place the number of weeks you wouldlike to see spent on each chapter. Each week consists of three lectures, one recitation, and a two-hourlaboratory. The total number of weeks should equal 24 since there is a week of introduction andorganization at the beginning of the quarter and a week of review at the end of the quarter. Please do notuse half-week units._____ Units, dimensions and vectors_____ Linear motion_____ Two dimensional motion_____ Forces and Newton's Laws_____ Applications of Newton's laws_____ Kinetic energy and work_____ Potential energy and conservation of energy_____ Momentum and collisions_____ Rotations and torque_____ Angular momentum_____ Statics_____ Gravitation_____ Simple harmonic motion_____ Waves (e.g. standing waves, sound, Doppler effect)_____ Superposition and interference of waves_____ Properties of fluids (e.g. pressure, continuity, Bernoulli's equation)_____ Temperature and ideal gas_____ Heat flow and the first law of thermodynamics_____ Molecules and gases (e.g. probability distributions of velocity, equipartition theory)_____ The second law of thermodynamics_____ Properties of solids (e.g. stress, strain, thermal expansion)_____ Electric charge (e.g. Coulomb's law, charge conservation)_____ Electric field_____ Gauss' law_____ Electric potential_____ Capacitors and dielectrics_____ Currents in materials (e.g. resistance, insulator, semiconductors)_____ DC circuits_____ Effects of magnetic fields (e.g. magnets, magnetic force, Hall effect)_____ Properties of magnetic fields (e.g. Ampere's law, Biot-Savart law)_____ Faraday's law_____ Magnetism and matter (e.g. ferromagnetism, diamagnetism)_____ Magnetic Inductance_____ AC circuits_____ Maxwell's equations and electromagnetic waves_____ Light (e.g. reflection and refraction)_____ Mirrors and lenses_____ Interference_____ Diffraction_____ Other. Please specify.24 Total number of weeksPlease place a star (*) next to the FOUR chapters listed above that you consider to be theMOST IMPORTANT for your students.

Department Survey 2375. The laboratory associated with this course is typically taught by graduate teaching assistants and could bestructured in several ways. Please place an 'X' by that structure that you feel would be most appropriate forthe students.___ A lab with well defined directions which verifies a physical principle previously explained to thestudents using the given apparatus.___ A lab where the students are given a specific question or problem for which they must conduct anexperiment with minimal guidance using the given apparatus.___ A lab where the students are given a general concept from which they must formulate an experimentalquestion, then design and conduct an experiment from a choice of apparatus.___ Other. Please describe.6. The recitation sections associated with this course is typically taught by graduate teaching assistants andcould be structured in several ways. Please place an 'X' by that structure that you feel would be mostappropriate for the students.___ Students ask the instructor to solve specific homework problems on the board.___ Instructor asks students to solve specific homework problems on the board.___ Instructor asks students to solve unfamiliar textbook problems, then discusses solution with class.___ Students work in small collaborative groups to solve real-world problems with the guidance of theinstructor.___ Other. Please describe.7. Would you please give examples of topics or subjects covered in your curriculum that assume someknowledge, skills or understanding which should be imparted by this physics course? Specific coursenumbers would also be helpful.Thank you for completing this questionnaire. If you have any material which illustrates the topics or subjectscovered in your curriculum which assume knowledge, skill, or understanding which should be imparted inPhysics, we would appreciate receiving a copy.In order for us to ask you more detailed questions and consult with you as the need arises, we ask that youcomplete the following information. Thank you.Name:Department / program:Campus address:Campus phone:

238 Appendix A

239Appendix BContext-rich Mechanics Problems1. (average speed) You and your friend run every day no matter what the weather (well almost). Onewinter day the temperature is a brisk 0 o F. Your friend insists that it is OK to run. You agree to thismadness as long as you both begin at your house and end the run at her nice warm house in a waythat neither of you has to wait in the cold. You know that she runs at a very consistent pace with anaverage speed of 3.0 m/s, while your average speed is a consistent 4.0 m/s. Your friend starts first sothat she will arrive at her house and unlock the door before you arrive. Five minutes later, you noticethat she dropped her keys and start after her. How far from your house will she be when you catchup if you run at your usual pace?2. (constant speed) You have been hired to assist in a research laboratory doing experimentsinvestigating the mechanism by which chemicals such as aspirin relieve pain. Your task is to calibrateyour detection equipment using the properties of a radioactive isotope that will later be used to trackthe chemical through the body. You have been told that your isotope decays by first emitting anelectron and then, some time later, emitting a photon. You set up your equipment to determine thetime between the electron emission and the photon emission. Your apparatus detects both electronsand photons. You determine that the electron and photon from a decay arrive at your detector at thesame time if the detector is 2.0 feet from your radioactive sample. A previous experiment has shownthat the electron from this decay travels at one half the speed of light. You know that the photontravels at the speed of light, 1.0 foot per nanosecond.3. (average speed) You have joined the University team racing a solar powered car. The optimalaverage speed for the car depends on the amount of sun hitting its solar panels. Your job is todetermine strategy by programming a computer to calculate the car's average speed for a dayconsisting of different race conditions. To do this you need to determine the equation for the day'saverage speed based on the car's average speed for each part of the trip. As practice you imagine thatthe day's race consists of some distance under bright sun, the same distance with partly cloudyconditions, and twice that distance under cloudy conditions.4. (average velocity) It is a beautiful weekend day and you and your friends decide to spend it outdoors.Two of your friends just want to relax while the other two want some exercise. You need some quiettime to study. To satisfy everyone, the group decides to spend the day on the river. Two people willput a canoe in the river and just drift downstream with the 1.5 mile per hour current. The secondpair will begin at the same time as the first but from 10 miles downstream. They will paddleupstream until the two canoes meet. Since you have been canoeing with these people before, youknow that they will have an average velocity of 2.5 miles per hour relative to the shore when they goagainst this river current. When the two canoes meet, they will come to shore and you should bethere to meet them with your van.5. (Constant velocity) You have a job in a University laboratory investigating possible accidentavoidance systems for oil tankers. Your group is concerned that an oil spill in the North Atlanticcould be caused by a super tanker running into an iceberg. The group has been developing a newtype of down-looking radar that can detect large icebergs. They are concerned about its rather shortrange of 2 miles. Super tankers are so huge that it takes a long time to turn them. Your job is todetermine how much time would be available to turn the tanker to avoid a collision once the icebergis detected. The radar signal travels at the speed of light, 186,000 miles per second, but once the

240 Appendix Bsignal arrives back at the ship it takes the computer 5 minutes to process it. A typical sailing speedfor super tankers during the winter on the North Atlantic is about 15 miles per hour. Assume thatthe tanker is heading directly at an iceberg that is drifting at 5 miles per hour in the same directionthat the tanker is going.6. (Constant velocity, group) You have a job in a research laboratory that has been investigatingpossible accident avoidance systems for automobiles. You have just begun a study of how bats avoidobstacles. In your study, a bat is fitted with a transceiver that broadcasts the bat's velocity to yourinstruments. Your research director has told you that the signal travels at the speed of light, 1.0ft/nanosecond. You know that the bat detects obstacles by emitting a forward going sound pulse(sonar) that travels at 1100 ft/s through the air. The bat detects the obstacle when the sound pulsereflects from the obstacle and that reflected pulse is heard by the bat. You are told to determine themaximum amount of time that a bat has after it detects the existence of an obstacle to change itsflight path to avoid the obstacle. In the experiment your instruments tell you that a bat is flyingstraight toward a wall at a constant velocity of 20.0 ft/s and emits a sound pulse when it is 10.0 ftfrom the wall.7. (1-D Kinematics) You are part of a citizen's group evaluating the safety of a high school athleticprogram. To help judge the diving program you would like to know how fast a diver hits the water inthe most complicated dive. The coach has the best diver perform for your group. The diver, jumpsfrom the high board, and near the end of the dive, passes the lower diving board, 3.0 m above thewater. Using a stopwatch, you determine that it took 0.20 seconds to enter the water from the timethe diver passed the lower board.8. (1-D Kinematics) You have a summer job working for a University research group investigating thecauses of the ozone depletion in the atmosphere. The plan is to collect data on the chemicalcomposition of the atmosphere as a function of the distance from the ground using a massspectrometer located in the nose cone of a rocket fired vertically. To make sure the delicateinstruments survive the launch, your task is to determine the acceleration of the rocket before it usesup its fuel. The rocket is launched straight up with a constant acceleration until the fuel is gone 30seconds later. To collect enough data, the total flight time must be 5.0 minutes before the rocketcrashes into the ground.9. (1-D Kinematics) Just for the fun of it, you and a friend decide to enter the famous Tour deMinnesota bicycle race from Rochester to Duluth and then to St. Paul. You are riding along at acomfortable speed of 20 mph when you see in your mirror that your friend is going to pass you atwhat you estimate to be a constant 30 mph. You will, of course, take up the challenge and acceleratejust as she passes you until you pass her. If you accelerate at a constant 0.25 miles per hour eachsecond until you pass her, how long will she be ahead of you?10. (1-D Kinematics, group) Because of your knowledge of physics, you have been assigned toinvestigate a train wreck between a fast moving passenger train and a slower moving freight trainboth going in the same direction. You have statements from the engineer of each train and thestationmaster as well as some measurements you make. To check the consistency the description ofevents leading up to the collision, you decide to calculate the distance from the station that thecollision should have occurred if everyone’s statement were accurate. You will then compare thatresult to the actual position of the wreck that is 0.5 miles from the station. In this calculation youdecide that you can ignore all reaction times. Here is what you know:

Context-rich Mechanics Problems 241a. • The stationmaster claims that she noted that the freight train was behind schedule. Asregulations require, she switched on a warning light just as the last car of the freight trainpassed her.b. • The freight train engineer says he was going at a constant speed of 10 miles per hour.c. • The passenger train engineer says she was going at the speed limit of 40 miles perhour when she approached the warning light. Just as she reached the warning light shesaw it go on and immediately hit the brakes.d. • The train reaches the warning light 2.0 miles before it gets to the station.e. • The passenger train's brakes slow it down at a constant rate of 1.0 mile per hour foreach minute they are applied.11. (1-D Kinematics, group) Because parents are concerned that children are learning "wrong" sciencefrom TV, you have been asked to be a technical advisor for a new science fiction show. The showtakes place on a space station at rest in deep space far away from any stars. In the plot, a viciouscriminal escapes from the space station prison. She steals a small space ship and blasts off to meether partners somewhere in deep space. Your job is to calculate how long her partners have totransport her off her ship before it is destroyed by a photon torpedo from the space station? In thestory, the stolen ship accelerates in a straight line at its maximum possible acceleration of 30 m/sec 2 .After 10 minutes all of the fuel is burned and the ship coasts at a constant velocity. Meanwhile, thehero learns of the escape and fires a photon torpedo. Once fired, a photon torpedo travels at aconstant velocity of 20,000 m/s. By that time the escapee has a 30 minute head start on the photontorpedo.12. (1 D Kinematics) You are an assistant technical advisor for a new gangster movie. In one scene therobbers try to evade capture by driving from one state to another. In the script, they are speedingdown the highway, and pass a concealed police car that is stopped by the roadside a short distancefrom the state line. The instant they pass the police car, it pulls onto the highway and acceleratesafter them at its maximum rate. The writers want to know how far this chase has to start from thestate line so that the robbers just make it across in dramatic style. The director wants you to give theanswer in terms of the constant speed of the gangster's car and the maximum acceleration of thepolice car sot the car's can be chosen.13. (1-D Kinematics, group) You have been hired as the assistant technical advisor for a new actionmovie. In this exciting scene, the hero pursues the villain up to the top of a bunge jumpingapparatus. The villain appears trapped but to create a diversion drops a bottle filled with a deadlynerve gas on the crowd below. The script calls for the hero to quickly strap the bunge cord to his legand dive straight down to grab the bottle while it is still in the air. Your job is to determine thelength of the unstretched bunge cord needed to make the stunt work. You estimate that the hero canjump off the bunge tower with a maximum velocity of 10 ft/sec. straight down by pushing off andcan react to the villain's dropping the bottle by strapping on the bunge cord and jumping in 2seconds.14. (1-D Kinematics, group) The University Skydiving Club has asked you to evaluate their plan a stuntfor an air show. The plan is for two skydivers to step out of opposite sides of a stationary hot airballoon 5,000 feet above the ground. The second skydiver will leave the balloon 20 seconds after thefirst skydiver but both are to land on the ground at the same time. To get a rough idea of thesituation, you decide to assume that a skydiver's air resistance will be negligible before the parachuteopens. As soon as the parachute is opened, the skydiver falls with a constant velocity of 10 ft/sec.The first skydiver waits 3 seconds after stepping out of the balloon before opening the parachute. To

242 Appendix Bdecide if this stunt will work, calculate how long the second skydiver waits after leaving the balloonbefore opening the parachute?15. (1-D Kinematics) You have a summer job as the technical assistant to the director of an adventuremovie. The script calls for a large package to be dropped onto the bed of a fast moving pick-uptruck from a helicopter that is hovering above the road. The helicopter is 235 feet above the road,and the bed of the truck is 3 feet above the road. The truck is traveling down the road at 40miles/hour. You must determine when to tell the helicopter to drop the package so it lands in thetruck.16. (1-D Kinematics, group) Because movie producers have come under pressure for teaching childrenincorrect science, you have been appointed to help a committee review a script for a new Supermanmovie. In the scene under consideration, Superman rushes to save Lois Lane who has been pushedfrom a window about 300 feet above a crowded street. Superman swoops down in the nick of time,arriving when Lois is half way to the street, and stopping her just at ground level. Lois changes herexpression from one of horror at her impending doom to a smile of gratitude as she gently floats tothe ground in Superman's arms. The committee wants to know if there is really enough time toexpress this range of emotions, even if there is a possible academy award on the line. The chairmanasks you to calculate the time it takes for Superman to stop Lois's fall. To do the calculation, youassume that Superman applies a constant force to Lois and that she weighs 110 lbs. While thinkingabout this scene you also wonder if Lois could survive the force that Superman applies to her so youcalculate that also.17. (2-D Kinematics) While on a vacation you find an old coastal fort probably built in the 16thcentury. Large stone walls rise vertically from the shore to protect the fort from cannon fire frompirate ships. Walking around the ramparts, you find the fort's cannons mounted such that they firehorizontally out of holes near the top of the walls facing the ocean. Leaning out of one of these gunholes, you drop a rock that hits the ocean 3.0 seconds later. You wonder if a ship 500 meters awaywould be in range of the fort's cannon? To answer that question you need to calculate how fast thecannon ball would have to leave the cannon to hit the ship.18. (2-D Kinematics) You read in the newspaper that rocks from Mars have been found on Earth. Yourfriend says that the rocks were shot off Mars by the large volcanoes there. You are skeptical so youdecide to calculate the magnitude of the velocity that volcanoes eject rocks from the geologicalevidence. You know the gravitational acceleration of objects falling near the surface of Mars is only40% that on the Earth. You assume that you can look up the height of Martian volcanoes and findsome evidence of the distance rocks from the volcano hit the ground from pictures of the Martiansurface. If you assume the rocks farthest from a volcano were ejected at an angle of 45 degrees, whatis the magnitude of the rock’s velocity as a function of its distance from the volcano and the heightof the volcano for the rock furthest from the volcano?19. (2-D Kinematics) You have a job on a team testing the software for an air traffic control center atthe nation’s busy airports. Part of the system is based on a single wide acceptance radar dish that candetect airplanes almost from horizon to horizon. The system determines an airplane’s averagevelocity from measurements at two different times. As a test you will have the aircraft fly directlyover the radar antenna on its downward going landing path. The system measures the distance of theairplane from the antenna and its angle from the horizontal at two times, before it passes over theantenna and after it has passed over the antenna. The distances, angles, and times are input into acomputer and the software calculates both the direction and magnitude of the airplane’s averagevelocity. You check the software by making the calculation.

Context-rich Mechanics Problems 24320. (2-D Kinematics) You have a summer job with an insurance company and have been asked to helpwith the investigation of a tragic "accident." When you visit the scene, you see a road runningstraight down a hill that has a slope of 10 degrees to the horizontal. At the bottom of the hill, theroad goes horizontally for a very short distance becoming a parking lot. The parking lot ends at thetop of a high cliff. Looking down from the edge of the parking lot, you see the wreck of the car onthe horizontal ground below the cliff. Police measurements give that the car is 30 feet from the baseof the cliff. The only witness claims that the car was parked on the hill, he can't exactly rememberwhere, and the car just began coasting down the road. He did not hear an engine so he thinks thatthe driver was drunk and passed out knocking off his emergency brake. He remembers that the cartook about 3 seconds to get down the hill. Your boss drops a stone from the edge of the cliff and,from the sound of it hitting the ground below, determines that it takes 5.0 seconds to fall to thebottom. After looking pensive, she tells you to calculate the car's average acceleration coming downthe hill based on the statement of the witness and the other facts in the case. She reminds you to becareful to write down all of your assumptions so she can evaluate the applicability of the calculationto this situation. Obviously, she suspects foul play.21. (2-D Kinematics) You have a summer job with an insurance company and have been asked to helpwith the investigation of a tragic "accident." When you visit the scene, you see a road runningstraight down a hill that has a slope of 10 degrees to the horizontal. At the bottom of the hill, theroad goes horizontally for a very short distance becoming a parking lot. The parking lot ends at thetop of a high cliff. Looking down from the edge of the parking lot, you see the wreck of the car onthe horizontal ground below the cliff. Police measurements give that the car is 30 feet from the baseof the cliff. Was it possible that the driver fell asleep at the wheel and simply drove over the cliff?After looking pensive, your boss tells you to calculate the speed of the car as it left the top of thecliff.22. (2-D Kinematics, group) Because of your physics background, you have been hired as a consultantfor a new movie about Galileo. In one scene, he climbs up to the top of a tower and, in frustrationover the people who ridicule his theories, throws a rock at a group of them standing on the ground.The rock leaves his hand at 30° from the horizontal. The script calls for the rock to land 15 m fromthe base of the tower near a group of his detractors. It is important for the script that the rock takesprecisely 3.0 seconds to hit the ground so that there is time for a good expressive close-up. The setcoordinator is concerned that the rock will hit the ground with too much speed causing cement chipsfrom the plaza to injure one of the high priced actors. You are told to calculate that speed.23. (2-D Kinematics) You are watching people practicing archery when you wonder how fast an arrow isshot from a bow. With a flash of insight you decide that you can easily determine what you want toknow by a simple measurement. You ask one of the archers to pull back the bowstring as far aspossible and shoot an arrow horizontally. The arrow strikes the ground at an angle of 86 degreesfrom the vertical at 100 feet from the archer's feet.24. (2-D Kinematics) You have a summer job working on the special effects team for a new movie. Abody is discovered in a field during the fall hunting season and the sheriff begins her investigation.One suspect is a hunter who was seen that morning shooting his rifle horizontally in the same field.He claims he was shooting at a deer and missed. You are to design the “flashback” scene that showshis version of firing the rifle with the bullet kicking up dirt where it hits the ground. The sheriff laterfinds a bullet in the ground. She tests the hunter’s rifle and finds the velocity that it shoots a bullet(muzzle velocity). In order to satisfy the nitpickers who demand that movies be realistic, the director

244 Appendix Bhas assigned you to calculate the distance from the hunter that this bullet should hit the ground as afunction of the bullet’s muzzle velocity and the rifle’s height above the ground.25. (2-D Kinematics) You have been hired as a member of a team investigating the ecosystemdevelopment around active volcanoes. One of your assignments is to write a computer program tohelp ensure the team's safety. You first decide to calculate the time a piece of volcanic material takesto reach an observing team. For this scenario, the team already knows the height of the volcanoabove them and their horizontal distance from the mouth of the volcano. They observe the anglethat the volcanic material leaves the mouth of the volcano.26. (2-D Kinematics) In your new job, you are helping to design stunts for a new movie. In one scenethe writers want a car to jump across a chasm between two cliffs. The car is driving along ahorizontal road when it goes over one cliff. Across the chasm, which is 1000 feet deep, is anotherroad at a lower height. They want to know the minimum speed of the car so that it does not fall intothe chasm. They have not yet selected the car so they want an expression for the speed of the car interms of the car's mass, the width of the chasm, and the height of the upper road above the lowerroad. They will plug in the actual numbers after they have purchased a car for the stunt.27. (2-D Kinematics) While you are watching a baseball game, the batter hits a ball that is barely off theground. The ball flies through the air and looks as if it is going to go over the fence 200 ft away for ahome run. In a spectacular play, the left fielder runs to the wall, leaps high, and catches the ball justas it clears the top of 10 ft high wall. You wonder how much time the fielder had to react to the hitand make the catch. You estimate that the ball left the bat at an angle of 30 o .28. (2-D Kinematics) You are a member of a citizen's committee investigating safety in the school sportsprogram. You are interested in knee damage to athletes participating in the long jump. The coachhas the best long jumper demonstrate the event for you. He runs down the track and, at the take-offpoint, jumps into the air at an angle of 30 degrees from the horizontal. He comes down in a sandpitat the same level as the track 8.5 m away from the take-off point. With what velocity (bothmagnitude and direction) did he hit the ground?29. (2-D Kinematics, group) Your friend has decided to make some money during the State Fair byinventing a game of skill for the Midway. In the game, the customer shoots a rifle at a 5.0 cmdiameter target 7.0 meters above the ground. At the instant that the bullet leaves the rifle, the targetis released from its holder and drops straight down. When shooting, the customer stands 20 metersfrom where the target would hit the ground if the bullet misses. Your friend asks you to calculatewhere to aim to hit the target in the center. The rifle's muzzle velocity is 350 m/sec according to themanual.30. (2-D Kinematics, group) Your group has been selected to serve on a citizen's panel to evaluate anew proposal to search for life on Mars. On this unmanned mission, the lander will leave orbitaround Mars falling through the atmosphere until it reaches 10,000 meters above the surface of theplanet. At that time a parachute opens and takes the lander down to 500 meters. Because of thepossibility of very strong winds near the surface, the parachute detaches from the lander at 500meters and the lander falls through the thin Martian atmosphere with a constant acceleration of 0.40gfor 1.0 second. Retrorockets then fire to bring the lander to a soft landing on the surface. A team ofbiologists has suggested that Martian life might be very fragile and decompose quickly in the heatfrom the lander. They suggest that any search for life should begin at least 9 meters from the base ofthe lander. This biology team has designed a probe that is shot from the lander by a springmechanism in the lander 2.0 meters above the surface of Mars. To return the data, the probe cannot

Context-rich Mechanics Problems 245be more than 11 meters from the bottom of the lander. For the probe to function properly it mustimpact the surface with a velocity of 8.0 m/s at an angle of 30 degrees from the vertical. Can thisprobe work as designed?31. (Circular kinematics) You have been hired as a consultant for a new Star Trek TV series to make surethat any science on the show is correct. In this episode, the crew discovers an abandoned spacestation in deep space far from any stars. This station is a large wheel-like structure where peoplelived and worked in the rim. In order to create "artificial gravity," the space station rotates on itsaxis. The special effects department wants to know at what rate a space station 200 meters indiameter would have to rotate to create "gravity" equal to 0.7 that at the surface of Earth.32. (Circular kinematics) While watching some TV you see a circus show in which a woman drives amotorcycle around the inside of a vertical ring. You determine that she goes around at a constantspeed and that it takes her 4.0 seconds to get around when she is going her slowest. If she is going atthe minimum speed for this stunt to work, the motorcycle is just barely touching the ring when she isupside down at the top. At that point she is in free fall. She just makes it around without falling offthe ring but what if she made a mistake and her motorcycle fell off at the top? How high up is she?33. (Forces, group) You have been asked to test a new device for precisely measuring the weight of smallobjects. The device consists of two light wires attached at one end to a support. An object isattached to the other end of each wire. The wires are far enough apart so the objects hanging onthem don’t touch. One of the objects has a very accurately known weight while the other object isthe unknown. A power supply is slowly turned on to give each object an electric charge causing themto slowly move away from each other because of the electric force. When the power supply is kept atits operating value, the objects come to rest at the same horizontal level. At that point, each of thewires supporting them makes a different angle with the vertical and that angle is measured. To testthe device, you calculate the weight of an unknown sphere from the measured angles and the weightof a known sphere. You will then check your calculation in the laboratory using a variety of differentobjects.34. (Forces, friction) You are driving your car uphill along a straight road. Suddenly, you see a car run ared light and enter the intersection just ahead of you. You slam on your brakes and skid in a straightline to a stop, leaving skid marks 100 feet long. A policeman observes the whole incident and gives aticket to the other car for running a red light. He also gives you a ticket for exceeding the speed limitof 30 mph (about 44 ft/s). When you get home, you read your physics book and estimate that thecoefficient of kinetic friction between your tires and the road was 0.60, and the coefficient of staticfriction was 0.80. You estimate that the hill made an angle of about 10° with the horizontal. Youlook in your owner's manual and find that your car weighs 2,050 lbs. Will you fight the traffic ticketin court?35. (Forces, friction, group) Because of your physics background, you have a job with a companydevising stunts for an upcoming adventure movie. In the script, the hero has just jumped on thetrain as it passed over a lake so he is wearing his rubber wet suit. He climbs up to the top of thelocomotive and begins fighting the villain while the train goes down a straight horizontal track at 5mph. During the fight, the hero slips and hangs by his fingers on the top edge of the front of thelocomotive. The locomotive has a smooth steel front face sloped at 20 o from the vertical so that thebottom of the front is more forward that the top. Now the villain stomps on the hero's fingers so hewill be forced to let go, slip down the front of the locomotive, and be crushed under its wheels.Meanwhile, the hero's partner is at the controls of the locomotive trying to stop the train. To add to

246 Appendix Bthe suspense, the brakes have been locked by the villain. It will take 7 seconds to open the lock. Tosave him, she immediately opens the throttle causing the train to accelerate forward such that thehero stays on the front face of the locomotive without slipping down. This gives her time to save hislife. The movie company wants to know the minimum acceleration necessary to perform this stunt.The hero weighs 180 lbs in his wet suit. The locomotive weighs 100 tons. You look in a book givingthe properties of materials and find that the coefficient of kinetic friction for rubber on steel is 0.50and its coefficient of static friction is 0.60.36. (Forces, friction) While working in a mechanical structures laboratory, your boss assigns you to testthe strength of ropes under different conditions. Your test set-up consists of two ropes attached to a30 kg block which slides on a 5.0 m long horizontal table top. Two low friction, light weight pulleysare mounted at opposite ends of the table. One rope is attached to each end of the 30 kg block.Each of these ropes runs horizontally over a different pulley. The other end of one of the ropes isattached to a 12 kg block which hangs straight down. The other end of the second rope is attachedto a 20 kg block also hanging straight down. The coefficient of kinetic friction between the block onthe table and the table's surface is 0.08. The 30 kg block is initially held in place by a mechanism thatis released when the test begins so that the block is accelerating during the test. During this test,what is the force exerted on the rope supporting the 12 kg block?37. (Forces, circular motion) After watching a TV show about Australia, you and some friends decide tomake a communications device invented by the Australian Aborigines. It consists of a noise-makerswung in a vertical circle on the end of a string. You are worried about whether the string you havewill be strong enough, so you decide to calculate the string tension when the device is swung with aconstant speed at a constant radius. You and your friends can't agree whether the maximum stringtension will occur when the noise maker is at the highest point in the circle, at the lowest point in thecircle, or is always the same. To settle the argument you decide to compare the tension at the highestpoint to that at the lowest point.38. (Forces, circular motion) One weekend, you decide to visit an amusement park and take yourneighbor's children. They want to ride on the 20 ft diameter Ferris wheel. The Ferris wheel has seatson the rim of a circle and rotates at a constant speed making one complete revolution every 20seconds. While you wait, you decide to calculate the total force (both magnitude and direction) onone of the children who weighs 45 lbs when they are at one quarter revolution past the highest point.Because the Ferris wheel can be run at different speeds, you also decide to make a graph that givesthe magnitude of the force on the child at that point as a function of the period of the Ferris wheel.39. (Forces, circular motion) You are reading a magazine article about the aesthetics of airplane design.One example in the article is a beautiful new design for commercial airliners. As a practical person,you want to know if the thin wing structure is strong enough to be safe. The article explains that anairplane can fly because the air exerts a force, called "lift," on the wings such that the lift is alwaysperpendicular to the wing surface. For level flying, the wings are horizontal. To turn, the pilot"banks" the plane so that the wings are oriented at an angle to the horizontal. This causes the planeto go in a horizontal circle. The specifications of the 100 x 10 3 lb plane require that it be able to turnwith a radius of 2.0 miles at a speed of 500 miles/hr. The article also states that tests show that thenew wing structure will support a force 4 times the lift necessary for level flight.

Context-rich Mechanics Problems 24740. (Forces, circular motion, group) You are flying to Chicago when the pilot tells you that the plane cannot land immediately because of airport delays and will have to circle the airport. You are told thatthe plane will maintain a speed of 450 mph at an altitude of 25,000 feet while traveling in a horizontalcircle around the airport. To pass the time you decide to figure out the radius of that circle. Younotice that to circle, the pilot "banks" the plane so that the wings are oriented at 10 o to thehorizontal. An article in your in-flight magazine explains that an airplane can fly because the airexerts a force, called "lift," on the wings. The lift is always perpendicular to the wing surface. Themagazine article gives the weight of the type of plane you are on as 100 x 10 3 pounds and the lengthof each wing as 150 feet. It gives no information on the thrust of the engines or the drag of theairframe.41. (Forces, circular motion, group) You are working with an ecology group investigating the feedinghabits of eagles. During this research, you observe an eagle circling in the air at a height that youestimate to be 300 feet. You suspect that the eagle is circling around its prey on the ground below.You wonder how far the eagle is from its prey so you decide to calculate it. Looking up at the eagle,you determine that its wings are banked to make an angle of 15 o from the horizontal and that it takes18 seconds to go around the circle.42. (Forces, circular motion) Your team is designing a package moving system in the new, improvedpost office. The device consists of a large circular disc (i.e. a turntable) that rotates once every 3.0seconds at a constant speed in the horizontal plane. Packages are put on the outer edge of theturntable on one side of the room and taken off on the opposite side. The coefficient of staticfriction between the disc surface and a package is 0.80 while the coefficient of kinetic friction is 0.60.If this system is to work, what is the maximum possible radius of the turntable?43. (Forces, circular motion) You were hired you as technical advisor for a new "James Bond" film. Thescene calls for Bond to chase a villain onto a merry-go-round. An accomplice starts the merry-goroundrotating in an effort to toss him off into an adjacent pool filled with hungry sharks. You mustdetermine a safe rate of rotation such that the stunt man will not fly off the merry-go-round and intothe pool. You measure the diameter of the merry-go-round to be 50 meters. You also determine thatthe coefficient of static friction between Bond's shoes and the merry-go-round surface is 0.7 and thecoefficient of kinetic friction is 0.5.44. (Forces, circular motion) You are driving with a friend who is sitting to your right on the passengerside of the front seat. The road you are on has some large turns in it. Without putting on a seatbeltyour friend might slide into you causing you to swerve and have an accident. When you stop forlunch, you try to convince your friend that this is possible. To do this you decide to calculate theminimum radius you could turn on a level road for your friend not to slide, based on the coefficientof static friction between your friend and the seat of the car and the car's constant speed. You alsodetermine the direction of the turn to make your friend slide into you and not away from you.45. (Forces, circular motion) You have been hired as a member team reviewing the safety of northernfreeways. The section of road you are studying has a curve that is 1/8 of a circle with a radius of 0.5miles. The road has been designed with a curve banked at an angle of 4 o to the horizontal. To beginthe study, you have been assigned to calculate the maximum speed for a standard passenger car(about 2000 lbs) to complete the turn while maintaining a horizontal path along the road if it iscovered by a slick coating of ice. You then need to compare your results to the case of a dry, clearroad with a coefficient of kinetic friction of 0.70 and coefficient of static friction of 0.80 between thetires and the road.

248 Appendix B46. (Forces, circular motion) On a trip through Florida, you find yourself driving in your along a flat levelroad at 50 mph. The road makes a turn that you take without changing speed. The curve isapproximately an arc of a circle with a radius of 0.05 miles. You notice that the road is flat and levelwith no sign of banking. There is no warning sign but you wonder if it would be safe to go 50 mpharound the curve in the rain when the wet surface has a lower coefficient of friction. To satisfy yourcuriosity, you decide to determine the minimum coefficient of static friction between the road andyour car's tires that will allow your car to make the turn. Your owner's manual says that your carweighs 3000-lbs.47. (Conservation of energy) You are driving your car uphill along a straight road. Suddenly, you see acar run a red light and enter the intersection just ahead of you. You slam on your brakes and skid ina straight line to a stop, leaving skid marks 100 feet long. A policeman observes the whole incidentand gives a ticket to the other car for running a red light. He also gives you a ticket for exceeding thespeed limit of 30 mph. When you get home, you read your physics book and estimate that thecoefficient of kinetic friction between your tires and the road was 0.60, and the coefficient of staticfriction was 0.80. You estimate that the hill made an angle of 10 o with the horizontal. In yourowner's manual you find that your car weighs 2,050 lbs. Will you fight the traffic ticket in court?48. (Conservation of energy) You are watching a National Geographic Special on television. Onesegment of the program is about the archer fish of Southeast Asia. This fish "shoots" water at insectsto knock them into the water so it can eat them. The commentator states that the archer fish keepsits mouth at the surface of the stream and squirts a jet of water from its mouth at 13 feet/second.You watch an archer fish shoot a moth off a leaf into the water. You estimate that the leaf was about2 feet above the stream. You wonder at what minimum angle from the horizontal the water can beejected from the fish's mouth to hit the moth. Since you have time during the commercial, youquickly calculate this angle.49. (Conservation of energy, force) At the train station, you notice a large horizontal spring at the end ofthe track where the train comes in. This is a safety device to stop the train so that it will not goplowing through the station if the engineer misjudges the stopping distance. While waiting, youwonder what would be the fastest train that the spring could stop by being fully compressed, 3.0 ft.To keep the passengers as safe as possible when the spring stops the train, you assume that themaximum stopping acceleration of the train, caused by the spring, is g/2. You make a guess that atrain might have a mass of 0.2 million kilograms. For the purpose of getting your answer, youassume that all frictional forces are negligible.50. (Conservation of energy) You are the technical advisor to a TV show about "death defying" stunts.Your task is to design a stunt in which a 5 ft 6 in, 120 pound actor jumps off a 100-foot tall towerwith an elastic cord tied to one ankle with the other end of the cord tied to the top of the tower.This 75 ft cord is very light but very strong and stretches so that it can stop the actor without pullinga leg off. Such a cord exerts a force with the same mathematical form as a spring. To minimize theforce that the cord exerts on the leg, you want it to stretch as far as possible. You must determinethe elastic force constant that characterizes the cord so that you can purchase it. For maximumdramatic effect, the jump will be off a diving board at the top of the tower. From tests you havemade, the maximum speed of a person coming off the diving board is 10 ft/sec.51. (Conservation of energy) You are the technical advisor to a TV show about "death defying" stunts.Your task is to design a stunt in which an 80 kg actor is shot out of a cannon elevated 40 o from the

Context-rich Mechanics Problems 249horizontal. The "cannon" is actually a 3-foot diameter tube that uses a stiff spring and a puff ofsmoke rather than an explosive to launch the human cannonball. The spring constant is 900Newtons/meter. The spring is compressed by a motor until its free end is 1.0 meters above theground. A small seat is attached to the free end of the spring for the actor to sit on. When thespring is released, it extends an additional 3 meters to the mouth of the tube. Neither the seat northe chair touch the sides of the tube. At the place that the actor would hit the ground you will put anairbag that exerts an average retarding force of 500 Newtons. You need to determine the minimumairbag thickness to stop the actor.52. (Conservation of energy, 2-D kinematics, group) You have a summer job with a company designingthe ski jump for the next Winter Olympics. The ski jump ramp is coated with snow and a skier glidesdown it reaching a high speed. The bottom of the ramp is shaped so that the skier leaves it travelinghorizontally. The winner is the person who jumps the farthest after leaving the end of the ramp.Your task is to write a computer program to determine the distance of the starting gate from thebottom of the ramp. For safety reasons, your design should be such that for a perfect run down theramp, the skier's speed before leaving the ramp and sailing through the air should not exceed a safespeed that will be input later. For a fixed angle of the ski ramp with the horizontal and a maximumsafe speed, your calculation should allow you to place the gate based on the skier's starting speed, thecoefficient of friction between the snow and the ski, and the mass of the skier. Since the ski-jumpersbend over and wear very aerodynamic suits, you decide to neglect the air resistance.53. (Conservation of momentum, kinematics) You have been hired as a technical consultant for an earlymorningcartoon series for children to make sure that the science is correct. In the script, a wagoncontaining two boxes of gold (total mass of 150 kg) has been cut loose from the horses by an outlaw.The wagon starts from 50 meters up a hill with a 6 o slope. The outlaw plans to have the wagon rolldown the hill and across the level ground and then crash into a canyon where the gang waits.Luckily, the Lone Ranger (mass 80 kg) and Tonto (mass 70 kg) are waiting in a tree 40 meters fromthe edge of the canyon. They drop vertically into the wagon as it passes beneath them. The scriptstates that it takes the Lone Ranger and Tonto 5.0 seconds to grab the gold and jump out of thewagon, but do they have that much time?54. (Conservation of momentum, kinematics) You have been asked to write part of the software for anew computer game. In one part of the game, the hero must get a magic fruit down from a tree byshooting it with an arrow. The hero aims the arrow so that when it is at the highest part of its path, ithits the fruit. The arrow sticks in the fruit and they both fall together to the ground. You need tocalculate the distance from the tree that the fruit hits the ground in terms of the speed that the arrowleaves the bow, the angle the arrow leaves the bow, the height of the tree, the mass of the arrow, andthe mass of the fruit.55. (Conservation of momentum) As a concerned citizen, you have volunteered to serve on a committeeinvestigating injuries to Junior High School students participating in sports programs. Currently thecommittee is looking into the high incidence of ankle injuries on the basketball team. You arewatching the team practice, looking for activities that can result in large horizontal forces on theankle. Observing the team practice jump shots gives you an idea, so you try a small calculation. A40-kg student jumps 0.6 meters straight up and shoots the 0.80-kg basketball at his highest point.From the trajectory of the basketball, you deduce that the ball left his hand at 30 o from the horizontalat 10 m/s. To help estimate the force, you calculate his horizontal velocity when he hits the ground?

250 Appendix B56. (Conservation of momentum) You are a volunteer at the Campus Museum of Natural History andhave been asked to assist in the production of an animated film about the survival of hawks in thewilderness. In the script, a 1.5-kg hawk is hovering in the air so it is stationary with respect to theground when it sees a goose flying below it. The hawk dives straight down at 60 km/hr. When itstrikes the goose it digs its claws into the goose's body. The goose, which has a mass of 2.5 kg, wasflying north at 30 km/hr just before it was struck by the hawk and killed instantly. The animatorswant to know the velocity (magnitude and direction) of the hawk and dead goose just after the strike.57. (Conservation of momentum, 2-D kinematics, or center of mass, group) While waiting in asupermarket line you read in a tabloid newspaper that an alien spaceship exploded while hoveringover the center of a remote town. The wreckage of the spaceship was found in three large pieces.One piece (mass = 300 kg) of the spaceship landed 6.0 km due north of the center of town. Anotherpiece (mass = 1000 kg) landed 1.6 km to the southeast (36 degrees south of east) of the center oftown. The last piece (mass = 400kg) landed 4.0 km to the southwest (65 degrees south of west) ofthe center of town. There were no more pieces of the spaceship. The paper reports that data fromair traffic control radar show the spaceship was hovering motionless over the center of town before itexploded and that just after the explosion the pieces initially moved horizontally. The articlespeculates that the spaceship did not explode on its own accord but was hit by a missile. When youget home you decide to determine whether the fragments reported are consistent with the spaceshipexploding spontaneously. If not, you want to know the direction from which the missile came.58. (Conservation of momentum, 2-D kinematics) While visiting a friend's house, you a fascinated bythe behavior of the cat. This 8 lb cat sits on the floor eyeing a stationary 20 lb chair that is 1.3 maway along the floor. The seat of the chair is 45 cm above the floor. The cat jumps up and lands onthe seat of the chair just as she reaches the maximum height of her trajectory. She puts out her clawsand hangs on. The chair sits on a part of the floor that has just been waxed and is very slippery. Asthe chair with the cat slides along the floor, you wonder what its speed was just after the cat lands onit?59. (Conservation of momentum) You have been asked to serve on a citizen’s commission investigatingthe safety of bridges across the Mississippi River. Because of increased barge traffic, you worry aboutbridge damage from being hit by runaway barges. A past accident serves as an example of what cango wrong. A fully loaded grain barge being pushed down river by tugboat rammed an empty bargebeing pushed directly across the river to a loading dock. At that time, the ropes tying each barge toits tugboat broke so that the barges were free. Records of the event give the speed of each barge andits mass before the accident. The record also gives the speed of the empty barge and its direction justafter the accident. Unfortunately the speed and direction of the loaded barge just after the accidentwas not recorded so you decide to calculate them.60. (Conservation of energy, conservation of momentum, group) Because of your physics background,you have been hired to help design stunt equipment for the movies. In this particular stunt, the actoris in a runaway car that skids in a complicated path down an icy hill. Your task is to design a methodof stopping the car at the bottom of the hill without harming the actor. To do this you decide tohave a steel plate at the bottom of the hill that the car can hit. The plate it attached to a stiff springwith the other end attached to a wall. The steel plate will be magnetized so that the car will stick toit. To choose the spring, you need to determine its spring constant based on the mass of the car, themass of the steel plate, the length of the spring, and the height that the car starts above the stoppingdevice.

Context-rich Mechanics Problems 25161. (Conservation of energy, conservation of momentum) You are helping a friend prepare for a skateboard exhibition. For the program, your friend plans to take a running start and then jump onto aheavy duty 15-lb stationary skateboard. The skateboard will glide in a straight line along a short, levelsection of track, then up a sloped concrete wall. The goal is to reach a height of at least 10 feetbefore turning to come back down the slope. You have measured your friend’s maximum runningspeed to safely jump on the skateboard at 7 feet/second. Can this program be carried out asplanned? Your friend weighs 120 lbs.62. (Conservation of energy, conservation of momentum) You have been hired as a technical advisor fora new James Bond movie. In the script, Bond and his latest love interest, who is 2/3 his weight(including skis, boots, clothes, and various hidden weapons), are skiing in the Swiss Alps. She skisdown a slope while he stays at the top to adjust his boot. When she has skied down a verticaldistance of 100 ft, she stops to wait for him and is captured by the bad guys. Bond looks up and seeswhat is happening. He notices that she is standing with her skis pointed downhill while she rests onher poles. To make as little noise as possible, Bond starts from rest and glides down the slopeheading right at her. Just before they collide, she sees him coming and lets go of her poles. He grabsher and they both continue downhill together. At the bottom of the hill, another slope goes uphilland they continue to glide up that slope until they reach the top of the hill and are safe. The writerswant you to calculate the maximum possible height that the second hill can be relative to the positionwhere he grabbed her. Both Bond and his girl friend are using new, top-secret frictionless stealth skisdeveloped for the British Secret Service.63. (Conservation of energy, conservation of momentum) You have been able to get a job with a medicalphysics group investigating ways to treat inoperable brain cancer. One form of cancer therapy beingstudied uses slow neutrons to deposit energy in the nucleus of the atoms that make up cancer cells.To test this idea, your research group decides to determine the change of energy of an iron nucleusafter a neutron knocks out one of its neutrons. In the experiment, a neutron goes into the nucleuswith a speed of 2.0 x 10 7 m/s and you detect two neutrons coming out at angles of 30 o and 15 o . To agood approximation, the heavy nucleus does not move during the interaction. The mass of a neutronis 1.7 x 10 -27 kg.64. (Conservation of energy, conservation of momentum) Your friend has just been in a traffic accidentand is trying to negotiate with the insurance company of the other driver to pay for fixing the car.You are asked to prove that it was the other driver's fault. At the scene of the crash you determinedwhat happened. Your friend was traveling North and entered an intersection with no stop sign. Atthe center of the intersection, this car was struck by the other car that was traveling East. The speedlimit on both roads entering the intersection is 50 mph. From the visible skid marks, the cars skiddedlocked together for 56 feet at an angle of 30 o north of east before stopping. At the library youdetermine that the weight of your friend's car was 2600 lbs and that of the other car was 2200 lbs,where you included the driver's weight in each case. The coefficient of kinetic friction for a rubbertire skidding on dry pavement is 0.80. It is not enough to prove that the other driver was speeding,you must also show that your friend was under the speed limit.65. (Conservation of energy, heat) You are helping a veterinarian friend to do some minor surgery on acow. Your job is to sterilize a scalpel and a hemostat by boiling them for 30 minutes. You do asordered and then quickly transfer the instruments to a well-insulated tray containing 200 grams ofsterilized water at room temperature which is just enough to cover the instruments. After a fewminutes the instruments and water will come to the same temperature, but can you hand to yourfriend without being burned? You know that both the 50 gram scalpel and the 70 gram hemostat are

252 Appendix Bmade from stainless steel that has a specific heat of 450 J / (kg o C). They were boiled in 2.0 kg ofwater with a specific heat of 4200 J / (kg o C).66. (Conservation of energy, heat) During the spring your friend will have an outdoor wedding. Youvolunteered to supply the perfect lemonade. Unfortunately, the ice used to cool the lemonade meltsdiluting it too much unless you have planned for this extra water. To help your planning, you lookup the specific heat capacity of water (1.0 cal/(gm o C)), the specific heat capacity of ice (0.50 cal/(gmoC)), and the latent heat of fusion of water (80 cal/gm). You assume that the specific heat capacity ofthe lemonade is the same as water. Using that information, you calculate how much water you getfrom the ice melting when you add just enough to 6 quarts (5.6 kg) of lemonade at room temperature(23 o C) to bring it down to 10 o C. The ice comes straight from the freezer at -5.0 ºC.67. (Conservation of energy, Conservation of momentum, Phase change) In a class demonstration, a 2.0-gram lead bullet was shot into a 2.0-kg block of wood hung from strings. The block of wood withthe bullet stuck in it swung up to a height 0.50 cm above its initial position. Is it possible that thebullet melts when it comes to rest in the block? Assume that the bullet had a temperature of 50 o Cwhen it left the gun. The melting temperature of lead is 330 o C. It has a specific heat capacity of 130J/(kg o C) and a latent heat of fusion of 25 J/g.68. (Conservation of energy, heat, pressure, group) While working for a mechanical engineeringcompany, your boss asks you to determine the efficiency of a new type of pneumatic elevator for usein a two level storage facility. The elevator is supported in a cylindrical shaft by a column of air,which you assume to be an ideal gas with a specific heat of 12.5 J/mol o C. The air pressure in thecolumn is 1.2 x 10 5 Pa when the elevator carries no load. The bottom of the cylindrical shaft isconnected to a reservoir of air at room temperature (25 o C). Seals around the elevator assure that noair escapes as the elevator moves up and down. The elevator has a cross-sectional area of 10 m 2 . Acycle of elevator use begins with the unloaded elevator. The elevator is loaded with 20,000 kgcausing the elevator to sink while the air temperature stays at 25 o C. The air in the shaft is thenheated to 75 o C and the elevator rises. The elevator is then unloaded, while the air remains at 75 o C.Finally, the air in the system is cooled to room temperature again, returning the elevator to its startinglevel. While the elevator is moving up and down, you assume that it moves at a constant velocity sothat the pressure in the gas is constant.69. (Center of mass, calculus, group) You have been hired as part of a research team consisting ofbiologists, computer scientists, engineers, mathematicians, and physicists investigating the virus thatcauses AIDS. This effort depends on the design of a new centrifuge to separate infected cells fromhealthy cells by spinning a container of these cells at very high speeds. Your design team has beenassigned the task of specifying the mechanical structure of the centrifuge arm that holds the samplecontainer. For aerodynamic stability, the arm must have uniform dimensions. Your team hasdecided the shape will be a long, thin rectangular strip. The properties of this strip will be optimizedby a computer program. The arm must be stronger at one end than at the other so your teamdecided to use new composite materials to accomplish this. With these materials one cancontinuously change the strength by changing the density of the arm along its length while keeping itsdimensions constant. The density will vary linearly along the length of the strip from a low value atits tip to a high value at its base. To calculate the strength of the brackets necessary to support thearm, you must determine the position of the center of mass of the arm as a function of thedimensions of the arm, its mass, the density at the tip, and the rate of change of the density along thearm.

Context-rich Mechanics Problems 25370. (Kinematics, rotation) You are working in a group investigating more energy efficient city busses.One option is to store energy in the rotation of a flywheel when the bus stops and then use it toaccelerate the bus. The flywheel under consideration is 1.5 m diameter disk of uniform constructionexcept that it has a massive, thin rim on its edge. Half the mass of the flywheel is in the rim. Whenthe bus stops, the flywheel rotates at 20 revolutions per second. When the bus is going at its normalspeed of 30 miles per hour, the flywheel rotates at 2 revolutions per second. The material holding therim to the rest of the flywheel has been tested to withstand an acceleration of up to 20g but you areworried that it might not be strong enough. To check, you consider the case of the bus initially going30 miles per hour making an emergency stop in 0.50 seconds. You assume that during this time theflywheel has a constant angular acceleration. You know that the moment of inertia of a disk rotatingabout its center is half that of a ring with the same mass and radius rotating about the same axis.71. (Conservation of energy, moment of inertia, rotation) You have applied for a summer job workingwith a special effects team at a movie studio. As part of your interview you have been asked toevaluate the design of a stunt in which a large spherical boulder rolls down an inclined track. At thebottom, the track curves up into a vertical circle so that the boulder can roll around on the inside ofthe circle and come back to ground level. It is important that the boulder not fall off the track andcrush the actors standing below. You have been asked to determine the relationship between theheight of the boulder's starting point on the ramp, as measured from the center of the boulder, andthe maximum radius the circular part of the track. You also know the properties of the boulder. Youare told that the moment of inertia of a sphere rotating about its center is 2/5 that of a ring with thesame mass and radius rotating about the same axis. After some thought you decide that the boulderwill stay moving in a vertical circle if its radial acceleration at the top is just that provided by gravity.72. (Torque, moment of inertia) After watching a news story about a fire in a high rise apartmentbuilding, you decide to design an emergency escape device from the top of a building. Because of thepossibility of power failure, you will make a gravitationally powered elevator. The design has a large,heavy horizontal disk that is free to rotate about its center on the roof with a cable wound around itsedge. The free end of the cable goes horizontally to the edge of the building roof, passes over aheavy vertical pulley, and then hangs straight down. A strong box that can hold 5 people is thenattached to the hanging end of the cable. When people enter the box and release the cable, it unrollsfrom the horizontal disk lowering the people safely to the ground. To see if this design is feasibleyou decide to calculate the acceleration of the fully loaded elevator to make sure it is much less thang. The device specifications are that the radius of the horizontal disk as 1.5 m and its mass is twicethat of the fully loaded elevator box. The disk that serves as the vertical pulley has 1/4 the radius ofthe horizontal disk and 1/16 its mass. You know that the moment of inertia of a disk about itscenter is 1/2 that of a ring with the same mass and radius about the same axis.73. (Torque, moment of inertia, kinematics) Because of your physics background, you have been askedto be a stunt consultant for a motion picture about a genetically synthesized prehistoric creature thatescapes from captivity and terrorizes the city. The scene you are asked to review has three actorschased by the creature through an old warehouse. At the exit of the warehouse is a thick steel firedoor 10 feet high and 6.0 feet wide weighing about 2,000 pounds. In the scene, the three actors fleefrom the building and close the fire door, thus sealing the creature inside. They have 3.0 seconds toshut the door and you need to know if they can do it. You estimate that each actor can each push onthe door with a force of 50 pounds. When they push together, each actor needs a space of about 1.5feet between them and the next actor. The door, with a moment of inertia around its hinges of 1/3of what it would be if all of its mass were concentrated at its outside edge, needs to rotate 120degrees to close.

254 Appendix B74. (Conservation of energy or torque, moment of inertia, kinematics) While working on a paper aboutthe technology of settlers crossing the Great Plains, you need to know the moment of inertia of awooden wagon wheel. You decide make a measurement on a wagon wheel from the museum. Thiswheel has a mass of 70 kg and a diameter of 1.3 m. You mount the wheel vertically on a low-frictionbearing then wrap a light cord around the outside of the rim to which you attach a 20-kg block.When the block is released, it falls 1.5 m in 0.33 s.75. (Conservation of energy or torque, moment of inertia, kinematics) While you watch a TV programabout life in the ancient world, you see that the people in one village used a solid sphere made out ofclay as a kind of pulley to help get water from a well. A well-greased wooden axle was placed throughthe center of the sphere and fixed in a horizontal orientation above the well, allowing the sphere torotate freely. To demonstrate the depth of the well, the host of the program completely wraps a thinrope around the sphere and ties the bucket to its end. When the bucket is released, it to falls to thebottom of the well unwinding the string from the rotating sphere as it goes. It takes 2.5 seconds.You wonder about the depth of the well so you decide to calculate it. You estimate that the spherehas twice the mass of the bucket. You look up the moment of inertia of a sphere about an axisthrough its center and find it is 2/5 that of a ring of the same mass and radius rotating about thesame axis.76. (Conservation of energy, torque, moment of inertia) You have been hired as a stunt advisor for amovie to be shot in Minnesota during the winter. The villain attempts to crush the hero by releasinga large sewer pipe from rest on a boat ramp. It rolls without slipping down the ramp and at thebottom of the ramp it encounters the horizontal, slick ice of a frozen stream. Having crossed thefrozen stream, the pipe starts up a second ramp that is covered with slick ice. The hero is standing atthe top of this ramp. The director wants the sewer pipe to almost reach her. Your assistant hasmeasured the maximum height (above the frozen stream) that the center of the sewer pipe can reach.He has also measured the mass and radius of the pipe and the different angles that the two rampsmake with the horizontal. You checked that frictional forces can be neglected on the slick ice. Atwhat height do you tell the crew to place the center of the sewer pipe before releasing it on the firstramp?77. (Conservation of energy, circular motion, moment of inertia) You are on a development teaminvestigating a new design for computer magnetic disk drives. You have been asked to determine ifthe standard disk drive motor will be sufficient for the test version of the new disk. To do this youdecide to calculate how much energy is needed to get the 6.4 cm diameter, 15 gram disk to itsoperating speed of 2000 revolutions per second. The test disk also has 4 different sensors attached toits surface. These small sensors are arranged at the corners of a square with sides of 1.2 cm. Toassure stability, the center of mass of the sensor array is in the same position as the center of mass ofthe disk. The disk’s axis of rotation also goes through the center of mass. You know that thesensors have masses of 1.0 grams, 1.5 grams, 2.0 grams, and 3.0 grams. The moment of inertia ofyour disk is one-half that of a ring of the same mass and radius rotating about the same axis.78. (Force, torque, center of mass) The automatic flag raising system on a horizontal flagpole attachedto the vertical outside wall of a tall building has become stuck. The management of the buildingwants to send a person crawling out along the flagpole to fix the problem. You have been askedwhether or not this is possible. The flagpole is a 120 lb steel I-beam that is very strong and rigid.One end of the flagpole is attached to the wall of the building by a hinge so that it can rotatevertically. Nine feet away, the other end of the flagpole is attached to a strong, lightweight cable.The cable goes up from the flagpole at an angle of 30 o until it reaches the building where it is bolted

Context-rich Mechanics Problems 255to the wall. The mechanic who will climb out on the flagpole weighs 150 lbs including equipment.From the specifications of the building construction, both the bolt attaching the cable to the buildingand the hinge have been tested to hold a force of 500 lbs. Your boss wants to know if the mechanicwill be ok at the far end of the flagpole, nine feet from the building.79. (Force, torque, moment of inertia) You have been asked to design a machine to move a large cablespool up a factory ramp at a constant speed. The spool is made of two 6.0 ft diameter disks of woodwith iron rims connected together at their centers by a solid cylinder 3.0 ft wide and 5.0 ft long.Sometime later in the manufacturing process, cable will be wound around the cylinder. For now thecylinder is bare but the spool still weighs 200 lbs. Your plan is to attach a light weight string aroundthe cylinder and pull the spool up the ramp with the string coming off the top of this cylinder. Thespool will then roll without slipping up the ramp on its two outside disks. To finish the design youneed to calculate how strong the string must be to pull the spool when it is moving up the ramp at aconstant speed. The ramp has an angle of 27 o from the horizontal and the string will be parallel tothe ramp.80. (Force, torque, kinematics) You are working with an archeological team reconstructing the techniquethat an ancient civilization used to move heavy stone blocks along level ground on a high mountainplateau. The theory you are testing claims that the ancients pulled a block along a greased woodenroad using a rope with one end attached to the block and the other end attached to a large rock. Thelarge rock was then dropped off a nearby cliff. To better control the motion of the block, the ropepassed over a large vertical wheel at the edge of the cliff supported by an axle through its center. Thewheel was free to rotate vertically. To recreate the situation, you need to determine the force that therope will exert on the block if it accelerates uniformly from rest and goes 150 m in 2.5 minutes. Youwill drop a 50 kg rock and use a wheel with a mass of 250 kg and a radius of 75 cm. The wheel isconstructed of a heavy iron rim that contains essentially all of its mass supported by light woodenspokes.81. (Force, torque, kinematics) While watching a mechanical clock mounted in a tower, you observe thatthe motion of the entire device is governed by a small lead ball hanging on one end of a string woundaround a large pulley. As the ball descends, the pulley rotates and the string unwinds without slippingon the pulley. You estimate that the mass of the lead ball is 125 grams and the mass of the pulley is2.5 kg. The pulley is shaped like a flat disc of radius 15 cm. You remember that the moment ofinertia of a disk is half that of a ring of the same mass and radius when they rotate about an axisthrough their center. If the ball starts from rest, you wonder how far it moves in 2 s?82. (Conservation of energy, rotation, moment of inertia, group) You are a member of a team designinga new device to help trucks go down steep mountain roads at a safe speed even if their brakes fail.The device is a flywheel, a large disk that rotates about its center and mounted in the truck. A systemof gears attaches the flywheel to the truck’s wheels so that when these gears are engaged, half of thetruck’s kinetic energy is in the flywheel. In order to design the flywheel, your team must know theforces on its structure. To help determine these forces, you decide to calculate the maximum radialand tangential components of the acceleration of any part of the flywheel when the truck starts fromthe top of a hill of a known height and goes down a straight road sloped at a known angle. The massof the truck and radius of the flywheel have already been determined for this design. The mass of theflywheel is 5% that of the loaded truck. The moment of inertia of the flywheel is one half that of thesame mass and radius ring rotating about its center. The flywheel gears can only be switched to theengaged position if both the truck and the flywheel are stopped.

256 Appendix B83. (Conservation of angular momentum, moment of inertia, calculus) You have been asked to helpevaluate a proposal to build a device to determine the speed of hockey pucks shot along the ice. Thedevice consists of a rod that rests on the ice and is fastened to the ice at one end so that it is free torotate horizontally. The free end of the rod has a small, light basket that will catch the hockey puck.The puck slides across the ice perpendicular to the rod and is caught in the basket. The rod thenrotates. The designers claim that knowing the mass of the rod and puck, the length of the rod, andthe rate of the rotation of the rod with the puck in the basket, you can calculate the speed of the puckas it moved across the ice before it hit the basket. To check their claim, you try to make thecalculation?84. (Conservation of energy, conservation of angular momentum, moment of inertia) You are helping todesign the opening ceremony for the next winter Olympics. One of the choreographers envisionsskaters racing out onto the ice and each one grabbing a very large ring (the symbol of the Olympics).Each ring is held horizontally at shoulder height by a vertical pole stuck into the ice. The pole isattached to the ring on its circumference so that the ring can rotate horizontally around the pole.The plan is to have a skater grab the ring at a point on the opposite side from where the pole isattached and, holding on, glide around the pole in a circle. You have been assigned the task ofdetermining the minimum speed that the skater must have before grabbing the ring in terms of theradius of the ring, the mass of the ring, the mass of the skater, and the constant frictional forcebetween the skates and the ice. The choreographer wants the skater and ring to go around the pole atleast five times. The skater moves tangent to the ring just before grabbing it.85. (Conservation of energy, conservation of angular momentum, moment of inertia) You have beenasked to design a new stunt for the opening of an ice show. A small 50 kg skater glides down a rampand along a short level stretch of ice. While gliding along the level stretch the skater bends to be assmall as possible finally grabbing the bottom end of a large 180 kg vertical rod that is free to turnvertically about a axis through its center. The plan is to hold onto the 20 foot long rod while itswings the skater to the top. You have been asked to give the minimum height of the ramp. Doing aquick integral tells you that the moment of inertia of this rod about its center is 1/3 of what itsmoment of inertia would be if all of its mass were concentrated at one of its ends.86. (Conservation of energy, conservation of angular momentum, moment of inertia, group) You are amember of a group designing an air filtration system for allergy suffers. To optimize its operationyou need to measure the mass of the common pollen in the air where the filter will be used. Youhave an idea of how to do this using a micro-machine. You imagine that a pollen particle enters anopening in your device. Once inside the pollen is given a positive electric charge and accelerated byan electrostatic force to a speed of 1.4 m/s. The pollen then hits the end of a very small, bar that ishanging straight down from a pivot at its top. Since the bar has a negative charge at its tip, the pollensticks to it as the bar swings up. Measuring the angle that the bar swings up would give the particle'smass. After the angle is measured, the charge of the bar is reversed, releasing that particle so that thedevice is ready for the next particle. Your friend insists it will never work and asks you to calculatethe length of the bar needed. You assume you want an angle of about 2 o for a typical pollen particleof 4 x 10 -9 grams. Your plan calls for a bar with a mass of 7x10 -4 grams and a moment of inertial 1/3as much as if all of its mass were concentrated at its end.87. (Conservation of angular momentum, Conservation of energy, moment of inertia) In the physics labyour group did not get the result you expected when a metal ring was dropped onto a rotating plate.After some thought you wonder if there could be a significant amount of friction on the rotatingshaft that would affect the result. You assume that this force is approximately constant, exceptperhaps just when the ring hits the disk. To measure that force, you drop the ring centered on the

Context-rich Mechanics Problems 257disk that you get spinning about its center at 3.0 revolutions per second. The disk and ring go around17 more times before coming to rest. The radii of the disk and shaft are 11 cm and 0.63 cm. Thering has an outside radius of 6.5 cm and inside radius of 5.5 cm. The moments of inertia (about theappropriate axis) for the disk, shaft, and ring are 5.1 x 10 -3 kg m 2 , 3.7 x 10 -6 kg m 2 , and 8.9 x 10 -3 kgm 2 respectively.88. (Gravitational force, circular motion, group) You found your physics course so interesting that youdecided to get a job working in a research group investigating the ozone depletion at the Earth'spoles. This group is planning to put an atmospheric measuring device in a satellite that will pass overboth poles. To collect samples of molecules escaping from the upper atmosphere, the satellite will beput into a circular orbit 150 miles above the surface of the Earth, which has a radius of about 4000miles. To adjust the instruments for the proper data taking rate, you need to calculate how manytimes per day the device will sample the atmosphere over the South pole. You do not remember thevalues of G or the mass of the Earth, but you do remember the acceleration with which objects fall atthe surface of the Earth.89. (Gravitational force, circular motion) While watching TV using your satellite receiver, you wonder ifthe satellite transmitting the signal stays over your town all of the time. If it does, how high must itbe?90. (Gravitational force, circular motion) As you notice a full moon rising, you wonder about thedistance from the earth to the moon. You know that it takes about 28 days for the moon to go oncearound the earth and that the radius of the earth is about 4000 miles. You do not remember theuniversal gravitational constant, G, but you do know the acceleration of an object dropped near thesurface of the earth.91. (Gravitational force, circular motion, group) You are reading a magazine article about a satellite inorbit around the Earth that detects X-rays coming from outer space. The article states that the X-raysignal detected from one source, Cygnus X-3, has an intensity that changes with a period of 4.8hours. This type of astronomical object emitting periodic signals is called a pulsar. One populartheory holds that the pulsar is a normal star (similar to our Sun) that orbits a much more massiveneutron star. The period of the X-ray signal is then the period of the orbit. The article claims thatthe distance between the normal star and the neutron star is approximately the same as the distancebetween the Earth and our Sun. You realize that if this is correct, you can determine how muchmore massive the Cygnus X-3 neutron star is than our Sun. You don't know the distance from theearth to the Sun or the value of the Universal Gravitational Constant, but you do remember theperiod of the earth around the Sun.92. (Gravitational force, Conservation of energy, conservation of angular momentum) You have a jobwith a research group investigating the destruction of the ozone layer in the atmosphere. They areplanning to orbit a satellite to monitor the amount of chlorine ions in the upper atmosphere overNorth America. It has been determined that the satellite should collect samples at a height of 100miles above the Earth's surface. At that height air resistance would make the amount of time thesatellite would stay in orbit too short to be useful so an elliptical orbit is planned. This orbit wouldallow the satellite to be close to the Earth over North America, where data was desired, but fartherfrom the Earth at a height of 1000 miles, out of most of the atmosphere, on the other side of ourplanet. To determine the apparatus needed to collect the data, you must calculate how fast thesatellite traveling at its lowest point? You do not remember the universal gravitational constant, G,but you do know that the radius of the earth is about 4000 miles.

258 Appendix B93. (Oscillations, Rotations) You are helping a friend build an experiment to test behavior modificationtechniques on rats. The design calls for an obstacle that swings across a path every second. To keepthe experiment as inexpensive as possible, you decide to use a meter stick as the swinging obstacle.Where do you drill a hole in the meter stick so that, when hanging by a nail through that hole, it willdo the job.94. (Oscillations, Rotations) Your friend is trying to construct a pendulum clock for a craft show andasks you for some advice. The pendulum will be a very thin, light wooden bar with a thin, but heavy,brass ring fastened to one end. The length of the rod is 80 cm and the diameter of the ring is 10 cm.For aesthetic reasons, the plan is to drill a hole in the bar to place the axis of rotation 15 cm fromone end. You have been asked to calculate the period of this pendulum.95. (Oscillations, Torque, Rotations) You have been asked to help design a system for applying resistivepaint to plastic sheeting to make containers that protect sensitive electronic components from electriccharges. The object used to apply the paint is a solid cylindrical roller. The roller is pushed back andforth over the plastic sheeting by a horizontal spring attached to a yoke that is attached to an axlethrough the center of the roller. The other end of the spring is attached to a fixed post. To apply thepaint evenly, the roller must roll without slipping over the surface of the plastic. In order todetermine how fast the process can proceed, you have been assigned to calculate how the oscillationfrequency of the roller depends on its mass, radius and the stiffness of the spring. You know that themoment of inertia of a solid cylinder with respect to an axis through its center is 1/2 that of a ringwith the same axis.96. (Oscillations, Conservation of Energy, Rotations, group) You have a job at a software company thatis producing a program simulating accidents in a modern commuter railroad station. Your task is todetermine the response of a safety system to prevent a railroad car from crashing into the station. Inthe simulation, a coupling fails causing a passenger car to break away from a train and roll into thestation. Because the brakes on the passenger car have failed, it cannot stop. The safety system at theend of the track is a large horizontal spring with a hook that will grab onto the car when it hitspreventing the car from crashing into the station platform. After the car hits the spring, yourprogram must calculate the frequency and amplitude of the car's oscillation based on thespecifications of the passenger car, the specifications of the spring, and the speed of the passengercar. In your simulation, the wheels of the car are disks with a significant mass and a moment ofinertia half that of a ring of the same mass and radius. At this stage of your simulation, you ignoreany energy dissipation in the car's axle or in the flexing of the spring, and the mass of the spring.97. (Waves) You have a summer job working on an oil tanker in the waters of Alaska. Your Captainknows that the ship is near an underwater ridge that could tear the bottom out. He estimates that itis about 6 km straight ahead of the ship. The ship's instruments tell him the ship is moving throughstill water at a speed of 31 km/hr but the captain cannot take any chances. He asks you to use thesonar to check the ship's speed. A sonar signal is sent out with a frequency of 980 Hz, bounces offthe underwater ridge, and is detected on the ship. If the ship's speed indicator is correct, whatfrequency should you detect? You use your trusty Physics text to find the speed of sound in seawateris 1522 m/s.98. (Waves) You've been hired as a technical consultant to the police department to design a detectorproofdevice that measures the speed of vehicles. You know that a moving car emits a variety ofcharacteristic sounds. You decide to make a very small device to be placed in the center of the roadthat will detect a specific frequency emitted by the car as it approaches and then measure the changein that frequency as the car moves off in the other direction. A microprocessor in the device will

Context-rich Mechanics Problems 259then compute the speed of the vehicle. To write the program for the microprocessor you need anequation for the speed of the car using the data received by the microprocessor. Your may alsoinclude necessary physical constants.99. (Standing waves) You have a job in a biomedical engineering laboratory working on technology toenhance hearing. You have learned that the human ear canal is essentially an air filled tubeapproximately 2.7 cm long that is open on one end and closed on the other. You wonder if there is aconnection between hearing sensitivity and the standing waves that can exist in the ear canal. To testyour idea, you calculate the lowest three frequencies of the standing waves in the ear canal. Fromyour Physics textbook, you find that the speed of sound in air is 343 m/s.

260 Appendix B

Appendix CContext-rich Electricity & Magnetism Problems2611. (Vector forces, Coulomb force, discrete charge, group) While working in a University researchlaboratory your group is given the job of testing an electrostatic scale, used to precisely measure theweight of small objects. The device consists of two light strings attached to a support so that theyhang straight down. A different object is attached to the other end of each string. One of theobjects has a very accurately known weight while the other object is the unknown. A power supply isslowly turned on to give each object an electric charge. This causes the objects to slowly move awayfrom each other. When the power supply is kept at its operating value, the objects come to rest atthe same horizontal level. At that time, each of the strings makes a different angle with the verticaland that angle is measured. To test your understanding of the device, you decide to calculate theweight of an unknown sphere from the measured angles and the weight of a known sphere. Yourknown is a standard sphere with a weight of 0.050 N supported by a string that makes an angle of10.00 o with the vertical. The unknown sphere's string makes an angle of 20.00 o with the vertical.2. (Coulomb force, discrete charge) While studying about the importance of hydrogen atoms in organicmolecules, you wonder about the energy states of a hydrogen atom. Using the planetary model of anatom, you decide to calculate the kinetic energy of the electron in a circular orbit around the protonas a function of the radius of the orbit and the properties of the electron and proton.3. (Coulomb force, discrete charge) You are working for a chemical company with a group trying toproduce new polymers. Your boss has asked you to help determine the structure of part of apolymer chain. She wants to know where a chlorine ion of effective charge -e would situate itselfnear a carbon dioxide ion. The carbon dioxide ion is composed of 2 oxygen ions each with aneffective charge -2e and a carbon ion with an effective charge +3e. You have been told to assumethat these ions are arranged in a line with the carbon ion sandwiched midway between the twooxygen ions. The distance between each oxygen ion and the carbon ion is 3.0 x 10 -11 m. Assumingthat the chlorine ion is on a line perpendicular to the axis of the carbon dioxide ion and that the linegoes through the carbon ion, where is its equilibrium position? For simplicity, you assume that thecarbon dioxide ion does not deform in the presence of the chlorine ion. Looking in your trustyphysics textbook, you find the charge of the electron is 1.60 x 10 -19 C4. (Coulomb force, continuous charge, group) You have a part time job in a research laboratory buildingequipment for experiments on the new space station. Because it is expensive to send heavyequipment into orbit, your group is investigating ideas for a lighter cathode ray tube. The design callsfor an electron acceleration mechanism consisting of two equal radius parallel rings separated bytwice their radius. Each ring is given an equal magnitude charge by a power supply. Electronsoriginate from a wire at the center of one of the rings and are accelerated along the axis between thecenters of the two rings. To estimate the variation of the electron's motion, you decide to calculatethe ratio of the electron's acceleration when it is at the center of one of the rings to its accelerationwhen it is midway between the two rings.5. (Electric field, continuous charges) You are helping to design a new electron microscope toinvestigate the structure of the HIV virus. A new device to position the electron beam consistsof a charged circle of conductor. This circle is divided into two half circles separated by a thininsulator so that half of the circle can be charged positively and half can be charged negatively.The electron beam will go through the center of the circle. To complete the design your job is to

262 Appendix Ccalculate the electric field in the center of the circle as a function of the amount of positivecharge on the half circle, the amount of negative charge on the half circle, and the radius of thecircle.6. (Electric field, continuous charges, calculus) You have a summer job with the telephone companyinvestigating the vulnerability of underground telephone lines to natural disasters. Your task is towrite a computer program that will be used determine the possible harm to a telephone wire from thehigh electric fields caused by lightning. The underground telephone wire is supported in the centerof a long, straight steel pipe that protects it. When lightening hits the ground it charges the steelpipe. Since you think that the largest field on the wire will be where it leaves the end of the pipe, youcalculate the electric field at that point as a function of the length of the pipe, the radius of the pipe,and the charge on the pipe.7. (Conservation of energy, Coulomb potential) While sitting in a restaurant with some friends, younotice that some "neon" signs are different in color than others. One of your friends, an art majorwho makes sculpture from these things, tells you that the color of the light depends on which gas isin the tube. All "neon" signs are not made using neon gas. You know that the color of light tells youits energy. Red light is a lower energy than blue light. Since the light is given off by the atoms thatmake up the gas, the different colors must depend on the structure of the gas atoms. Another one ofyour friends has read about the Bohr theory that states electrons are in uniform circular motionaround a heavy, motionless nucleus in the center of the atom. This theory also states that theelectrons are only allowed to have certain orbits. When an atom changes from one allowed orbit to alower one, it radiates light as required by the conservation of energy. Since only certain orbits areallowed, only light of certain energies (colors) can be emitted. You decide to explore the theory bycalculating the energy of light emitted by a hydrogen atom when an electron makes a transition fromone allowed orbit to another. You remember that the proton has a mass 2000 times that of anelectron. When you get home you look in your textbook and find the electron mass is 9 x 10 -31 kgand its charge is 1.6 x 10 -19 C. The radius of the smallest allowed electron orbit for hydrogen is 0.5 x10 -10 meters. The next allowed orbit has a radius 4 times as large as the smallest orbit.8. (Conservation of energy, Coulomb potential) You have a job working in a cancer research laboratory.Your team is trying to construct a gas laser that will give off light of an energy that will pass throughthe skin but be absorbed by cancer tissue. You know that an atom emits a photon (light) when anelectron goes from a higher energy orbit to a lower energy orbit. Only certain orbits are allowed in aparticular atom. To begin the process, you calculate the energy of photons emitted by a Helium ionin which the electron changes from an orbit with a radius of 0.40 nanometers to another orbit with aradius of 0.26 nanometers.9. (Conservation of energy, Coulomb potential) You have been asked to write operating instructionsfor a new device that deposits ions on the surface of silicon to make better semiconductors. Thedevice accelerates the ions in a straight line from rest as they pass through a potential differencebetween two conductive parallel plates. The plates have a hole through them to let the ions gothrough. After leaving the parallel plates, the ions travel a long distance to the silicon. The smallpiece of silicon is at the center of circle with six small electrodes with the same charge at equaldistances around the circumference. The plane of the circle is perpendicular to the ion beam. Theentire device is sealed so that the ion travels in a vacuum. You need to determine the relationshipbetween the charge on each electrode and the potential difference between the two parallel plates sothat the ion stops on the silicon surface. You may need to know the distance of each electrode from

Context-rich Electricity & Magnetism Problems 263the silicon, the distance of each electrode to the next electrode, the charge of the ion, the mass of theion, and the distance between the parallel plates and the silicon.10. (Conservation of energy, Coulomb force, continuous charge, group, calculus) You have been askedto determine if a proposed apparatus to implant ions in silicon to make better semiconductors willwork. The apparatus slows down positive He ions that have a charge twice that of an electron(He ++ ). It consists of a circular wire that is connected to a power supply so that it becomes anegatively charged circle. An ion with a velocity of 200 m/s on a trajectory perpendicular to theplane of the circle is shot out from the center of the circle. The wire circle has a radius of 3.0 cm andcan have a charge up to 8.0 µC. The sample into which the ion is to be implanted is to be placed 2.5mm from the charged circle. You look up the charge of an electron and mass of the helium and findthem to be 1.6 x 10 -19 C and 6.7 x 10 -27 Kg.11. (Conservation of energy, Coulomb potential, continuous charge, calculus) You have been hired by acompany engaged in developing faster computer chips by implanting certain ions into the silicon thatmakes up the chips. Your job is to help design a new device to do this. You start with a thin bar ofsilicon that is given a uniform positive charge. The device will then direct negative ions from an ionsource along a path aligned with the axis of the silicon bar so that the ions hit the end of the bar. Tocontrol the process, you need to know the acceleration of the ion as a function of its distance fromthe end of the silicon rod that the ions hit when you are given the properties of the ion and thecharge and size of the silicon rod.12. (Conservation of energy, Coulomb potential, group) You have a job in a University laboratory that isplanning experiments to study the forces between nuclei in order to understand the energy output ofthe Sun. In one of these experiments, alpha particles are shot from a Van de Graaf accelerator at asheet of lead. The alpha particle is the nucleus of a helium atom and is made of 2 protons and 2neutrons. The lead nucleus is made of 82 protons and 125 neutrons. The mass of the neutron isalmost the same as the mass of a proton. You have been told to calculate the potential differencebetween the two ends of the Van de Graaf accelerator so that the alpha particle should come intocontact with a lead nucleus. The alpha particle has a radius of 1.0 x 10 -13 cm and the lead nucleus hasa radius 4 times larger.13. (Conservation of energy, Coulomb potential, Gravitational potential) NASA has asked your team ofrocket scientists about the feasibility of a new satellite launcher that will save rocket fuel. NASA'sidea is basically an electric slingshot that consists of 4 electrodes arranged in a horizontal square 5 mon a side. The satellite is placed 15 m directly under the center of the square. A power supply willprovide each of the four electrodes with the same charge and the satellite with an opposite charge 4times larger. When the satellite is released from rest, it moves up and passes through the center ofthe square. At the instant it reaches the square's center, the power supply is turned off and theelectrodes are grounded, giving them a zero electric charge. To test this idea, you decide to useenergy considerations to calculate the electrode charge necessary to get a 100 kg satellite to an orbitheight of 300 km. In your physics text you find the mass of the Earth to be 6.0 x 10 24 kg.14. (Conservation of energy, Coulomb potential, continuous charge, calculus) You have been asked toevaluate a design for accelerating electrons in a precise electron microscope. Pulses of electrons areshot into the center of a 3.0 cm diameter wire ring along the axis perpendicular to the plane of thering. When an electron reaches the center of the ring, the ring is rapidly charged to 4.5 mC. Youwant to know the speed of the electron when it is very far away from the ring if it starts with a speed

264 Appendix Cof 6.0 m/s at the center. From you Physics book you find the charge of an electron is 1.6 x 10 -19 Cand its mass is 9.1 x 10 -31 kg.15. (Conservation of energy, Coulomb potential, continuous charge, calculus, group) You have beenasked to evaluate a new electron gun design for producing low velocity electrons. The electrons havea 20 cm path from the heating element that emits them to the end of the gun. The electrons mustreach the end of the gun with a speed of 10 2 m/s. After leaving the heating element, the electronspass through a 5.0 mm diameter hole in the center of a 3.0 cm diameter charged circular disk. Thedisk's positive charge is kept at 3.0 µC/m 2 . The heating element is a spherical electrode 0.10 mm indiameter whose negative charge can be adjusted up to -0.10 mC. There is 1.0 cm between the heatingelement and the hole in the disk. Your co-worker thinks that the hole in the disk is too large toignore in your calculations. Using your physics text you find that the mass of the electron is 9.11 x10 -31 kg and its charge is 1.6 x 10 -19 C.16. (Coulomb potential, Electric force) You've been hired to design the hardware for an ink jet printer.Your printer uses a deflecting electrode to cause charged ink drops to form letters on a page.Uniform ink drops of about 30 microns radius are charged while being sprayed out towards the pageat a speed of about 20 m/s. Along the way to the page, they pass into a region between twodeflecting plates that are 1.6 cm long. The deflecting plates are 1.0 mm apart and charged to 1500volts. You measure the distance from the edge of the plates to the paper and find that it is one-halfinch. Assuming an uncharged droplet forms the bottom of the letter, how much charge is needed onthe droplet to form the top of a letter 3 mm high17. (Gauss' Law, conservation of energy, Coulomb potential, calculus) You are working in cooperationwith the Public Health Department to design a device to measure particles from auto emissions. Theaverage particle has a mass of 6.0 x 10 -9 kg. When it enters the device it is exposed to ultravioletradiation that knocks off electrons so that it has a charge of +3.0 x 10 -8 C. This average particle isthen moving at a speed of 900 m/s and is 15 cm from a very long negatively charged wire with alinear charge density of -8.0 x 10 -6 C/m. The detector for the particle is located 7.0 cm from thewire. In order to design the proper kind of detector, you need to know the speed that an averageemission particle hits the detector. They tell you that an average emission particle has a mass of 6.0 x10 -9 kg.18. (Conservation of energy, Coulomb potential, Gauss' Law, calculus) You have a summer job in aresearch laboratory with a group investigating the possibility of producing power from fusion. Thedevice being designed confines a hot gas of positively charged ions, called plasma, in a very longcylinder with a radius of 2.0 cm. The charge density of the plasma in the cylinder is 6.0 x 10 -5 C/m 3 .Positively charged Tritium ions are to be injected into the plasma perpendicular to the axis of thecylinder in a direction toward the center of the cylinder. Your job is to determine the speed that aTritium ion should have when it enters the plasma cylinder so that its velocity is zero as it reaches theaxis of the cylinder. Tritium is an isotope of Hydrogen with one proton and two neutrons. You lookup the charge of a proton and the mass of tritium in your Physics text to be 1.6 x 10 -19 C and 5.0 x10 -27 Kg.19. (Capacitance, potential, Gauss' Law, group, calculus) Your team is designing an inexpensiveemergency electrical system for a rural hospital. Someone has suggested storing energy in largecapacitors made from two thin walled metal pipes. The pipes would be concentric and of differentradii but the same length. The idea is to first charge the capacitor by connecting each pipe to theopposite terminals of a power supply. After each pipe has its maximum charge, the power supply is

Context-rich Electricity & Magnetism Problems 265disconnected. One of the engineers on the team has an idea to increase the energy stored in acapacitor by changing its capacitance after it is charged. After the power supply is disconnected, amechanical device would insert a third concentric metal thin walled pipe between the original two butnot touching them. To evaluate the usefulness of this idea, you decide to calculate the ratio of thecapacitance of the final three pipe configuration to that of the original two pipe configuration as afunction of the size of the pipes.20. (Capacitance) As part of your summer job as a design engineer at an electronics company, youhave been asked to evaluate the circuit shown below to determine whether a dangerous amountof energy is stored in the capacitors.50 F9V60 F100 F21. (Resistance, Ohm's law) You have a summer job as an assistant technician for a telephone companyin California. During a recent earthquake, a 1.0-mile long underground telephone line is crushed atsome point. This telephone line is made up of two parallel copper wires of the same diameter andsame length, which are normally not connected. At the place where the line is crushed, the two wiresmake contact. Your boss wants you to find this place so that the wire can be dug up and fixed. Youdisconnect the line from the telephone system by disconnecting both wires of the line at both ends.You then go to one end of the line and connect one terminal of a 6.0-V battery to one wire, and theother terminal of the battery to one terminal of an ammeter. When the other terminal of theammeter is connected to the other wire, the ammeter shows that the current through the wire is 1.0A. You then disconnect everything and travel to the other end of the telephone line, where yourepeat the process and find a current of 1/3 A.22. (Conservation of energy, power) You are working with a company to design a new, 700-foot high,50-story office building. The owner has discovered that it would take the 6500-lb loaded elevatorone minute to rise 20 stories and thinks this is too long for these busy executives to spend in anelevator. You have been told to speed up the elevator and decide to buy a bigger power supply for it.You find a supply that it is the same as the old one except that it outputs twice the voltage. Now youmust calculate the operating expenses of the new power supply. You estimate that while the elevatorruns at maximum speed, the whole system, including the power supply, is 60% efficient. The cost ofelectricity is $0.06 per kilowatt-hour.23. (Conservation of energy, power) You are having some friends over for dinner and decide to cookspaghetti. You start by boiling 5.0 kg of water. To decide when to start the water, you need toestimate how long it will take to get to its boiling point. Your electric stove is a 200-ohm device that

266 Appendix Coperates at 120 volts. Checking your physics book you find that water has a specific heat capacity of4200 J/(kg o C) and heat of vaporization is 2.3 x 106 J/kg.24. x(Circuits, power, Ohm's law) You are working in a group designing electronics for use in anunderground neutrino experiment. One of your concerns is that the power dissipated by the circuitsince will cause the laboratory temperature to increase affecting the performance of the apparatus andthe comfort of the people. You have determined that most of the problem comes from the part ofthe circuit shown below. You decide to calculate the total power dissipated by all of the resistorstogether as well as the power dissipated by the 6 ohm resistor.12 V326425. x(Circuits, power, Ohm's law, group) You are on a safety team reviewing a large chemical plant. Toincrease the speed of a chemical reaction, a tank of potentially explosive liquid is heated using thesystem shown in the diagram below. A 400 ohm heating element's temperature is controlled by theother three resistors in the circuit. The power is supplied by a 1200 volt power supply. To protectagainst excessive heating, the heating element and the 200 ohm resistor are monitored by infraredsensors. If something goes dangerously wrong, the ratio of the power dissipated by these resistorswill become large. If that ratio goes over 10,000 an alarm will sound, all power is cut off, and thefactory is evacuated. To determine the sensitivity of this safety system, you are asked to calculate thatratio during normal operations.400 200 1200 V 100 300 26. x(Circuits, Ohm's law, group) You have been asked to check a circuit that will become part of a newcommunications satellite to see if it will function as intended. This circuit contains batteries that willpower the satellite when it is in the Earth’s shadow. When the satellite is in the light, its solar panelsare intended to charge up the batteries by supplying a constant current at points a and b as shown inthe circuit below. You know the properties of the two identical batteries and the identical resistors.

Context-rich Electricity & Magnetism Problems 267IabI+ +27. x(Circuits, Ohm's law, group) You are on a team designing a new electric car. The amount ofcurrent that the battery supplies to the electric motor controls the speed of the car. To make asimple current control, you connect a resistive wire in series with the battery. One terminal of themotor is then connected to one terminal of the battery. The other terminal of the motor isconnected to a point on the resistive wire that can be adjusted. This connection divides the resistivewire into two resistors whose ratio can be changed by moving the point of contact on the resistivewire. To see if you can get enough power from this arrangement, you decide to determine how thecurrent through the motor depends on the properties of the battery and the electric motor, the totalresistance of the resistive wire, and the ratio of resistances into which it is divided.28. x(Circuits, power, Ohm's law) You have been assign to determine the rate that batteries run downwhen operating a new portable amplifier for cell phones. You have analyzed that part of the circuitand find it is equivalent to the one below consisting of three identical resistors and two identicalbatteries. You decide to calculate the rate that each battery runs down as a function of the propertiesof the batteries and the resistors.29. x(Circuits, power, Ohm's law) As part of your summer job at an electronics company, you havebeen asked to evaluate the circuit shown below. The resistors are rated at 0.5 Watts, whichmeans they burn-up if their power output exceeds 0.5 Watts. Is the 100W resistor safe?100 9 V 133 6 V200 30. (Magnetic force, electric potential) You decide to relax by watching some TV. After a few minutes,you are bored and your mind starts drifting. You think about how the TV picture is generated. In atypical color picture tube for a TV, the electrons start from a cathode at the back of the tube. Nearthat cathode the electrons are accelerated through about 20,000 volts then go at a constant speed tohit the picture tube screen about 1.5 ft away. On the screen is a grid of color dots about 1/100 inchapart. When electrons hits one of them, the dot glows the appropriate color producing the color

268 Appendix Cpicture. You know that the Earth's magnetic field is different at different places in the world andwonder if the TV's location has an effect on the picture. You remember that a typical value for theEarth's magnetic field is 0.5 Gauss. In your Physics text you find that the mass of a proton is 1.67 x10 -27 kg and its charge is 1.6 x 10 -19 C.31. (Magnetic force, group) You have a summer job in the medical school research group investigatingshort lived radioactive isotopes to use in fighting cancer. Your group is working on a way oftransporting alpha particles (Helium nuclei) from their source to another room where they will collidewith other material to form the isotopes. Your job is to design that part of the transport system thatwill deflect the beam of alpha particles (mass of 6.64 x 10 -27 kg, charge of 3.2 x 10 -19 C) through anangle of 90 o by using a magnetic field. The beam will be traveling horizontally in an evacuated tube.To make the alpha particles turn 90 degrees, you decide to use a dipole magnet that provides auniform vertical magnetic field of 0.030 T. Your design has a tube of the appropriate shape betweenthe poles of the magnet. Before you submit your design for consideration, you must determine howlong the alpha particles will spend in the uniform magnetic field while making the 90 o -turn.32. (Magnetic force) Your team has been told to develop a method for keeping charged particles fromdamaging the Space Telescope's sensitive electronic equipment. The Space Telescope is essentially ahollow cylinder of diameter of 2.0 meters with a cover on one end. Any particle entering parallel tothe axis of the telescope could damage the equipment. Your plan is to turn on a uniform magneticfield outside the end of the telescope whenever its cover is opened. The design goal is for a 0.010Tesla field to deflect a particle with a charge of 79 protons, a mass 197 times that of a proton, and avelocity of 1.0 x 10 6 m /s from a path down the axis of the cylinder to hit the wall at an angle of 20degrees. Assuming no magnetic field enters the telescope, how far from the telescope opening mustthe magnetic field extend into space to achieve this goal? In your Physics text you find that the massof a proton is 1.67 x 10 -27 kg and its charge is 1.6 x 10 -19 C.33. (Magnetic force, calculus, group) You have been asked to evaluate the design for a new electric trainthat uses the Earth's magnetic field to propel it. A current goes through one rail of the track, upthrough one of the train's wheels, through the wheel's horizontal axle, through the wheel at the otherend of the axle, and then through the other rail of the track. To check the feasibility of this design,you decide to calculate the speed of the train as a function of the mass of the train, the currentthrough the rails, the length of the axle, the radius of the wheels, and the time elapsed since the trainstarted up. You also look up the magnitude of the vertical component of the Earth's magnetic field.34. (Magnetic force, Ampere's Law, calculus) You have been asked to review a design for an electronbeam device for making electronic microcircuits. During part of its trajectory, the electron beamruns parallel to a section of a long wire at a distance of 5.0 cm from the wire. To determine theeffect on the beam, you decide to calculate the force on an electron when a current of 40 amps isswitched through the wire. You know that the mass of an electron is 9.1 x 10 -31 kg, and it is movingat 3.0 x 10 7 m/s.35. (Magnetic force, Ampere's Law, calculus, group) You are designing the supports for a high voltagepower line to bring electricity to the city from a dam. The two copper cables comprising the powerline will run side by side. Each cable hangs from light-weight vertical non-conductive straps, 80 cmlong, attached to concrete poles. Before current is turned on, the straps hang straight downsupporting the cables so that they are separated by 10 cm. For structural reasons, the cables cannotbe separated by more than 15 cm or less than 5 cm. You need to specify the maximum current thatyour design will allow. The copper cables that will be used have a weight of 100 N per meter.

Context-rich Electricity & Magnetism Problems 26936. (Biot-Savart Law, calculus) You are continually having troubles with the CRT screen of yourcomputer and wonder if it is due to magnetic fields from the power lines running in your building. Ablueprint of the building shows that the nearest power line is as shown below. Your CRT screen islocated at point P. You decide to calculate the magnetic field at P as a function of the current I andthe distances a and b. Segments BC and AD are arcs of concentric circles. Segments AB and DC arestraight line segments.37. (Magnetic force, Biot-Savart Law, torque, calculus) You are investigating ways to exert torque onparts of nanomachines. One possible device is a large multi-turn circular coil of wire that conducts acurrent. At the center of that large coil, a very small wire square that is mounted so that it rotatesabout one of its sides. The current in the small coil will be different than the current in the large coil.Determine the maximum torque on the square in terms of the size and number of turns of the largecoil, the size of the wire square, and the current through each.38. (Hall Effect, or Magnetic force, Electric force, potential, calculus, group) You are helping to design anew device to measure the blood flow through arteries as a diagnostic device for heart surgery. Theidea is to use microsurgery to attach small wires to opposite sides of an artery. The wires are broughtoutside of the body and attached to a voltmeter. Helmholz coils outside the body are then used tocreate a uniform magnetic field at the artery that is perpendicular to the blood flow and to thedirection between the wire attachment points. After the magnetic field is turned on, the ions in theblood come to equilibrium such that there is a constant electric field in the artery. You decide tocalculate the smallest velocity of blood flow that this device can measure for an artery of diameter 3mm in a magnetic field of 0.5 T if the voltmeter is accurate to 1 microvolt.39. (Magnetic force, Faraday's law, Ohm's law, calculus, group) You have a summer job working at acompany developing systems to safely move large loads down ramps. The safety system your team isinvestigating consists of a conducting bar sliding on two parallel conducting rails that run down theramp. The bar is perpendicular to the rails and is in contact with them. At the bottom of the ramp,the two rails are connected together. A magnet with poles above and below the ramp creates avertical uniform magnetic field. Before setting up a laboratory test, you decide to calculate thevelocity of the bar sliding down the ramp as a function of the mass of the bar, the strength of themagnetic field, the angle of the ramp from the horizontal, the distance between the tracks, and theresistance of the bar. Assume that all of the other conductors in the system have a much smallerresistance than the bar.

270 Appendix C40. (LC circuit, calculus) You are evaluating the design of a circuit to use a coil of wire to generate amagnetic field. In this circuit, a coil of wire, a capacitor, and a battery with known properties areconnected together in series. You worry how the current through the coil varies with time. Forsimplicity, you assume that the coil and all wires in the circuit have a negligible resistance.

271APPENDIX DPROBLEM SOLVING LABORATORIESThe following are examples of laboratories that emphasize problemsolving integrated with the measurement of physical systems. Theseexamples are taken from the first year physics course.LABORATORY I:DESCRIPTION OF MOTION IN ONE DIMENSIONIn this laboratory you will study the motion of objects that can go in only onedimension; that is, along a straight line. You will be able to measure the positionof these objects by capturing video images on a computer. Through thesemeasurements and their analysis you can investigate the relationship of quantitiesthat are useful to describe the motion of objects. Determining these kinematicquantities (position, time, velocity, and acceleration) under different conditionsallows you to improve your intuition about their quantitative relationships. Inparticular, you should be able to determine which relationships depend onspecific situations and which apply to all situations.There are many possibilities for the motion of an object. It might either move ata constant speed, speed up, slow down, or have some combination of thesemotions. In making measurements in the real world, you need to be able toquickly understand your data so that you can tell if your results make sense toyou. If your results don't make sense, then either you have not set up thesituation properly to explore the physics you desire, you are making yourmeasurements incorrectly, or your ideas about the behavior of objects in thephysical world are incorrect. In any of the above cases, it is a waste of time tocontinue making measurements. You must stop, determine what is wrong andfix it. If it is your ideas that are wrong, this is the time to correct them bydiscussing the inconsistencies with your partners, rereading your text, or talkingwith your instructor. Remember, one of the reasons for doing physics in alaboratory setting is to help you confront and overcome your incorrectpreconceptions about physics, measurements, calculations, and technicalcommunications. Because most people are much faster at recognizing patternsin pictures than in numbers, the computer will graph your data as you go along.

272OBJECTIVES:After you successfully complete this laboratory, you should be able to:Describe completely the motion of any object moving in one dimensionusing the concepts of position, time, velocity, and acceleration.Distinguish between average quantities and instantaneous quantities whendescribing the motion of an object.Express mathematically the relationships among position, time, velocity,average velocity, acceleration, and average acceleration for differentsituations.Graphically analyze the motion of an object.Begin using technical communication skills such as keeping a laboratoryjournal and writing a laboratory report.PREPARATION:Read Tipler: Chapter 2. Also read Appendix D, the instructions for doing videoanalysis. Before coming to the lab you should be able to:Define and recognize the differences among these concepts:- Position, displacement, and distance.- Instantaneous velocity and average velocity.- Instantaneous acceleration and average acceleration.Find the slope and intercept of a straight-line graph. If you need help, seeAppendix C.Determine the slope of a curve at any point on that curve. If you need help,see Appendix C.Determine the derivative of a quantity from the appropriate graph.Use the definitions of sin , cos , and tan with a right triangle.

273PROBLEM #1:CONSTANT VELOCITY MOTIONThese laboratory instructions may be unlike any you have seen before.You will not find worksheets or step-by-step instructions. Instead, eachlaboratory consists of a set of problems that you solve before comingto the laboratory by making an organized set of decisions (a problemsolving strategy) based on your initial knowledge. The instructions aredesigned to help you examine your thoughts about physics. These labsare your opportunity to compare your ideas about what "should"happen with what really happens. The labs will have little value inhelping you learn physics unless you take time to predict what willhappen before you do something. While in the laboratory, take yourtime and try to answer all the questions in this lab manual. In particular,the exploration questions are important to answer before you makemeasurements. Make sure you complete the laboratory problem,including all analysis and conclusions, before moving on to the next one.Since this design may be new to you, this first problem contains boththe instructions to explore constant velocity motion and an explanationof the various parts of the instructions. The explanation of theinstructions is in this font and is preceded by the double, vertical linesseen to the left.Why are we doing this lab problem? How is it related to the real world?In the lab instructions, the first paragraphs describe a possible situationthat raises the problem you are about to solve. This emphasizes theapplication of physics in solving real-world problems.To earn some extra money, you have taken a job as a camera operator forthe Minneapolis Grand Prix automobile race. Since the race will besimulcast on the Internet, you will be using a digital video camera thatstores the images directly on a computer. You notice that the image isdistorted near the edges of the picture and wonder if this affects themeasurement of a car’s speed from the video image. You decide tomodel the situation using a toy car, which moves at a constant velocity.?Does the measured speed of a car moving with aconstant velocity depend on the position of the car in avideo picture?The question, framed in a box and preceded by a question mark, definesthe experimental problem you are trying to solve. You should keep thequestion in mind as you work through the problem.

274EQUIPMENTTo make a prediction about what you expect to happen, you need tohave a general understanding of the apparatus you will use before youbegin. This section contains a brief description of the apparatus and thekind of measurements you can make to solve the laboratory problem.The details should become clear to you as you use the equipment.For this problem, you will use a motorized toy car, which moves with aconstant velocity on an aluminum track. You will also have a stopwatch,a meter stick, a video camera and a computer with a video analysisapplication written in LabVIEW (described in Appendix D) to helpyou analyze the motion. In the computer the LabVIEW applicationprograms include VIDEOPLAYER and VIDEOTOOL.PREDICTIONEveryone has his/her own "personal theories" about the way the worldworks. One purpose of this lab is to help you clarify your conceptionsof the physical world by testing the predictions of your personal theoryagainst what really happens. For this reason, you will always predictwhat will happen before collecting and analyzing the data. Yourprediction should be completed and written in your lab journalbefore you come to lab. The “Method Questions” in the next sectionare designed to help you determine your prediction and should also becompleted before you come to lab. This may seem a little backwards.Although the prediction question is given before the methodquestions, you should complete the method questions beforemaking the prediction. The prediction question is given first so youknow your goal.Spend the first few minutes at the beginning of the lab sessioncomparing your prediction with those of your partners. Discuss thereasons for any differences in opinion. It is not necessary that yourpredictions are correct, but it is necessary that you understand the basisof your prediction.How would each of the graphs of position-versus-time, velocity-versustime,and acceleration-versus-time show a distortion of the positionmeasurement? Sketch these graphs to illustrate your answer. Howwould you determine the speed of the car from each of the graphs?Which method would be the most sensitive technique for determiningany distortions? Appendix B might help you answer this question.

275Sometimes, as with this problem, your prediction is an "educated guess"based on your knowledge of the physical world. There is no way tocalculate an exact answer to this problem. For other problems, you willbe asked to use your knowledge of the concepts and principles ofphysics to calculate a mathematical relationship between quantities in theexperimental problem.WARM UP QUESTIONSMethod Questions are a series of questions intended to help you solvethe experimental problem. They either help you make the prediction orhelp you plan how to analyze data. Warm Up Questions should beanswered and written in your lab journal before you come to lab.To determine if the measured speed is affected by distortion, you need tothink about how to measure and represent the motion of an object. Thefollowing questions should help with the analysis of your data.1. How would you expect an instantaneous velocity-versus-time graph to lookfor an object moving with a constant velocity? Make a rough sketchand explain your reasoning. Write down the equation that describesthis graph. If this equation has any constant quantities in it, what arethe units of those constant quantities? What parts of the motion ofthe object does each of these constant quantities represent? For a toycar, what do you estimate should be the magnitude of thosequantities? How would a distortion affect this graph? How would itaffect the equation that describes the graph? How will theuncertainty of your position measurements affect this graph? Howmight you tell the difference between uncertainty and distortion?2. How would you expect a position-versus-time graph to look for an objectmoving with a constant velocity? Make a rough sketch and explainyour reasoning. What is the relationship between this graph and theinstantaneous velocity versus time graph? Write down the equationthat describes this graph. If this equation has any constant quantitiesin it, what are the units of those constant quantities? What parts ofthe motion of the object does each of these constant quantitiesrepresent? For a toy car, what do you estimate should be themagnitude of those quantities? How would a distortion affect thisgraph? How would it affect the equation that describes the graph?How will the uncertainty of your position measurements affect thisgraph? How might you tell the difference between uncertainty anddistortion?

2763. How would you expect an instantaneous acceleration-versus-time graph tolook for an object moving with a constant velocity? Make a roughsketch and explain your reasoning. Write down the equation thatdescribes this graph. If this equation has any constant quantities in it,what are the units of those constant quantities? What parts of themotion of the object does each of these constant quantitiesrepresent? For a toy car, what do you estimate should be themagnitude of those quantities? How would a distortion affect thisgraph? How would it affect the equation that describes the graph?How will the uncertainty of your position measurements affect thisgraph? How might you tell the difference?EXPLORATIONThis section is extremely important—many instructions will not make sense,or you may be lead astray, if you do not take the time to carefullyexplore your experimental plan.In this section you practice with the apparatus before you make timeconsumingmeasurements which may not be valid. This is where youcarefully observe the behavior of your physical system, before you beginmaking measurements. You will also need to explore the range overwhich your apparatus is reliable. Remember to always treat theapparatus with care and respect. Your fellow students in the next labsection will need to use the equipment after you are finished with it. Ifyou are unsure about how the apparatus works, ask your lab instructor.Most apparatus has a range in which its operation is simple andstraightforward. This is its range of reliability. Outside of that range,complicated corrections need to be applied. You can quickly determinethe range of reliability by making qualitative observations at what youconsider to be the extreme ranges of your measurements. Record yourobservations in your lab journal. If you observe that the apparatus doesnot function properly for the range of quantities you were consideringmeasuring, you can modify your experimental plan before you havewasted time taking an invalid set of measurements.The result of the exploration should be a plan for doing themeasurements that you need. Record your measurement plan inyour journal.Place one of the metal tracks on your lab bench and place the toy car onthe track. Turn on the car and observe its motion. Determine if itactually moves with a constant velocity. Use the meter stick andstopwatch to determine the speed of the car.

277Turn on the video camera and look at the motion as seen by the cameraon the computer screen. Go to Appendix D for instructions about usingthe video recorder.Do you need to focus the camera to get a clean image? How do theroom lights affect the image? Which controls help sharpen the image?Record your camera adjustments in your lab journal.Move the position of the camera closer to the car. How does this affectthe video image on the screen? Try moving it farther away. Raise theheight of the camera tripod. How does this affect the image? Decidewhere you want to place the camera to minimize the distortion.Practice taking a video of the toy car. Looking at the video frame byframe allows you to check whether the computer has missed any frames(the motion should be smooth). The capacity of the computer to take inall of the data from the video camera depends on the amount of data. Ifyour computer is dropping too many frames, you will not have enoughdata to analyze. You can minimize the number of frames dropped bydecreasing the amount of data in the video picture by adjusting thepicture size and keeping the picture as feature free as possible. Check outthese effects. Write down the best situation for taking a video in yourjournal for future reference. You will be doing a lot of this. When youhave the best movie possible, save it and open the video analysisapplication.Make sure everyone in your group gets the chance to operate the camera and thecomputer.MEASUREMENTNow that you have predicted the result of your measurement and haveexplored how your apparatus behaves, you are ready to make carefulmeasurements. To avoid wasting time and effort, make the minimalmeasurements necessary to convince yourself and others that you havesolved the laboratory problem.Measure the speed of the car using a stopwatch as it travels a knowndistance. How many measurements should you take to determine thecar’s speed? (Too few measurements may not be convincing to others,too many and you may waste time and effort.) How much accuracy doyou need from your meter stick and stopwatch to determine a speed to atleast two significant figures? Make the number of measurements youneed and record them in a neat and organized manner so that you canunderstand them a month from now if you must. Also make sure torecord precisely how you make these measurements. Some future labproblems will require results from earlier ones.

278Take a video of the motion of the car to determine its speed. Measuresome object you can see in the video so that you can tell the analysisprogram the real size of the video images when it asks you to calibrate.The best object to measure is the car itself. When you digitize the video,why is it important to click on the same point on the car’s image?Estimate your accuracy in doing so. Be sure to take measurements of themotion of the car in the distorted regions (edges) of the video.Make sure you set the scale for the axes of your graph so that you can seethe data points as you take them. Use your measurements of totaldistance the car travels and total time to determine the maximum andminimum value for each axis before taking data.Are any points missing from the position versus time graph? Missingpoints result from more data being transmitted from the camera than thecomputer can write to its memory. If too many points are missing, makesure that the size of your video frame is optimal (see Appendix D). Itmay also be that your background is too busy. Try positioning yourapparatus so that the background has fewer visual features.Note: Be sure to record your measurements with the appropriate number of significantfigures (see Appendix A) and with your estimated uncertainty (see Appendix B).Otherwise, the data is nearly meaningless.ANALYSISData by itself is of very limited use. Most interesting quantities are thosederived from the data, not direct measurements themselves. Yourpredictions may be qualitatively correct but quantitatively very wrong.To see this you must process your data.Always complete your data processing (analysis) before you take yournext set of data. If something is going wrong, you shouldn't waste timetaking a lot of useless data. After analyzing the first data, you may needto modify your measurement plan and re-do the measurements. If youdo, be sure to record the changes in your plan in your journal.Calculate the average speed of the car from your stopwatch and meterstick measurements. Determine if the speed is constant within yourmeasurement uncertainties. Can you determine the instantaneous speedof your car as a function of time?Analyze your video to find the instantaneous speed of the car as afunction of time. Determine if the speed is constant within yourmeasurement uncertainties. See Appendix D for instructions on how todo video analysis.

279Why do you have less data points for the velocity versus time graphcompared to the position versus time graph? Use the data tablesgenerated by the computer to explain how the computer generates thevelocity graphs.CONCLUSIONSAfter you have analyzed your data, you are ready to answer theexperimental problem. State your result in the most general termssupported by your analysis. This should all be recorded in yourjournal in one place before moving on to the next problemassigned by your lab instructor. Make sure you compare yourresult to your prediction.Compare the car’s speed measured with video analysis to themeasurement using a stopwatch. How do they compare? Did yourmeasurements and graphs agree with your answers to the MethodQuestions? If not, why? What are the limitations on the accuracy ofyour measurements and analysis?Do measurements near the edges of the video give the same speed as thatas found in the center of the image within the uncertainties of yourmeasurement? What will you do for future measurements?

280PROBLEM #3:MOTION UP AND DOWN AN INCLINEA proposed ride at the Valley Fair amusement park launches a rollercoaster car up an inclined track. Near the top of the track, the carreverses direction and rolls backwards into the station. As a member ofthe safety committee, you have been asked to compute the accelerationof the car throughout the ride. To check your results, you decide to builda laboratory model of the ride.?What is the acceleration of an object moving up anddown a ramp at all times during its motion?EQUIPMENTFor this problem you will have a stopwatch, a meter stick, an end stop, awood block, a video camera and a computer with a video analysisapplication written in LabVIEW. You will also have a cart to roll upthe inclined track.PREDICTIONMake a rough sketch of how you expect the acceleration-versus-timegraph to look for a cart given an initial velocity up along an inclinedtrack. The graph should be for the entire motion of going up the track,reaching the highest point, and then coming down the track.Do you think the acceleration of the cart moving up an inclined track will be greater than, lessthan, or the same as the acceleration of the cart moving down the track? What is the accelerationof the cart at its highest point? Explain your reasoning.WARM UP QUESTIONSThe following questions should help with your prediction and theanalysis of your data.1. Sketch a graph of how you expect an instantaneous acceleration-versus-timegraph to look if the cart moved down the track with the direction of aconstant acceleration always down along the track. Sketch a graph ofhow you expect an instantaneous acceleration-versus-time graph to look ifthe cart moved up the track with the direction of a constant

281acceleration always down along the track after an initial push. Whenthe cart moves up the track and then down the track, graph howwould you expect an instantaneous acceleration-versus-time graph to lookfor the entire motion after an initial push? Explain your reasoningfor each graph. To make the comparison easier, it is useful to drawthese graphs next to each other. Write down the equation(s) that bestrepresents each of these graphs. If there are constants in yourequation, what kinematic quantities do they represent? How wouldyou determine these constants from your graph?2. Write down the relationship between the acceleration and the velocityof the cart. Use that relationship to construct an instantaneousvelocity versus time graph just below each of your acceleration versustime graphs from question 1. The connection between the derivativeof a function and the slope of its graph will be useful. Use the samescale for your time axes. Write down the equation that best representseach of these graphs. If there are constants in your equation, whatkinematic quantities do they represent? How would you determinethese constants from your graph? Can any of these constants bedetermined from the constants in the equation representing theacceleration versus time graphs? Which graph do you think bestrepresents how velocity of the cart changes with time? Change yourprediction if necessary.3. Write down the relationship between the velocity and the position ofthe cart. Use that relationship to construct a position versus timegraph just below each of your velocity versus time graphs fromquestion 2. The connection between the derivative of a function andthe slope of its graph will be useful. Use the same scale for your timeaxes. Write down the equation that best represents each of thesegraphs. If there are constants in your equation, what kinematicquantities do they represent? How would you determine theseconstants from your graph? Can any of these constants bedetermined from the constants in the equation representing thevelocity versus time graphs? Which graph do you think bestrepresents how position of the cart changes with time? Change yourprediction if necessary.EXPLORATIONWhat is the best way to change the angle of the inclined track in areproducible way? How are you going to measure this angle with respectto the table? Think about trigonometry. How steep of an incline do youwant to use?Start the cart up the track with a gentle push. BE SURE TO CATCHTHE CART BEFORE IT HITS THE END STOP ON ITS WAYDOWN! Observe the cart as it moves up the inclined track. At the

282instant the cart reverses direction, what is its velocity? Its acceleration?Observe the cart as it moves down the inclined track. Do yourobservations agree with your prediction? If not, this is a good time tochange your prediction.Where is the best place to put the camera? Is it important to have mostof the motion in the center of the picture? Which part of the motion doyou wish to capture?Try several different angles. Be sure to catch the cart before itcollides with the end stop at the bottom of the track. If the angle istoo large, the cart may not go up very far and give you too few videoframes for the measurement. If the angle is too small it will be difficultto measure the acceleration. Determine the useful range of angles foryour track. Take a few practice videos and play them back to make sureyou have captured the motion you want.Choose the angle that gives you the best video record.What is the total distance through which the cart rolls? How much timedoes it take? These measurements will help you set up the graphs foryour computer data taking.Write down your measurement plan.MEASUREMENTUsing the plan you devised in the exploration section, make a video ofthe cart moving up and then down the track at your chosen angle. Makesure you get enough points for each part of the motion to determine thebehavior of the acceleration. Don't forget to measure and record the angle (withestimated uncertainty).Choose an object in your picture for calibration. Choose your coordinatesystem. Is a rotated coordinate system the easiest to use in this case?Why is it important to click on the same point on the car’s image torecord its position? Estimate your accuracy in doing so.Make sure you set the scale for the axes of your graph so that you can seethe data points as you take them. Use your measurements of totaldistance the cart travels and total time to determine the maximum andminimum value for each axis before taking data.Are any points missing from the position versus time graph? Missingpoints result from more data being transmitted from the camera than the

283computer can write to its memory. If too many points are missing, makesure that the size of your video frame is optimal (see Appendix D). Itmay also be that your background is too busy. Try positioning yourapparatus so that the background has fewer visual features.ANALYSISChoose a function to represent the position versus time graph. How canyou estimate the values of the constants of the function from the graph?You can waste a lot of time if you just try to guess the constants. Whatkinematic quantities do these constants represent? Can you tell fromyour graph where the cart reaches its highest point?Choose a function to represent the velocity versus time graph. How canyou calculate the values of the constants of this function from thefunction representing the position versus time graph? Check how wellthis works. You can also estimate the values of the constants from thegraph. Just trying to guess the constants can waste a lot of your time.What kinematic quantities do these constants represent? Can you tellfrom your graph where the cart reaches its highest point?From the velocity versus time graph determine if the accelerationchanges as the cart goes up and then down the ramp. Use the functionrepresenting the velocity versus time graph to calculate the accelerationof the cart as a function of time. Make a graph of that function. Can youtell from your graph where the cart reaches its highest point? Is theaverage acceleration of the cart equal to its instantaneous acceleration inthis case?As you analyze your video, make sure everyone in your group gets the chance tooperate the computer.CONCLUSIONHow do your position-versus-time and velocity-versus-time graphscompare with your answers to the method questions and the prediction?What are the limitations on the accuracy of your measurements andanalysis?Did the cart have the same acceleration throughout its motion? Did theacceleration change direction? Was the acceleration zero at the top of itsmotion? Did the acceleration change direction? Describe theacceleration of the cart through its entire motion after the initial push.

284Justify your answer. What are the limitations on the accuracy of yourmeasurements and analysis?

285 CHECK YOUR UNDERSTANDING1. Suppose you are looking down from a helicopter at three cars traveling in the samedirection along the freeway. The positions of the three cars every 2 seconds arerepresented by dots on the diagram below. The positive direction is to the right.Car At1t2t3t4t5t6Car Bt 1t 2t 3t 4t 5Car Ct 1t 2t 3t 4t5t6a. At what clock reading (or time interval) do Car A and Car B have very nearly thesame speed? Explain your reasoning.b. At approximately what clock reading (or readings) does one car pass another car? Ineach instance you cite, indicate which car, A, B or C, is doing the overtaking.Explain your reasoning.c. Suppose you calculated the average velocity for Car B between t 1 and t 5 . Where wasthe car when its instantaneous velocity was equal to its average velocity? Explainyour reasoning.d. Which graph below best represents the position-versus-time graph of Car A? Of CarB? Of car C? Explain your reasoning.e. Which graph below best represents the instantaneous velocity-versus-time graph ofCar A? Of Car B? Of car C? Explain your reasoning. (HINT: Examine thedistances traveled in successive time intervals.)Which graph below best represents the instantaneous acceleration-versus-time graphof Car A? Of Car B? Of car C? Explain your reasoning.+++(a)t(b)t(c)t+++(d)t(e)t(f)t+++ttt-(g)-(h)-(i)

2862. A cart starts from rest at the top of a hill, rolls down the hill, over a short flat section,then back up another hill, as shown in the diagram above. Assume that the frictionbetween the wheels and the rails is negligible. The positive direction is to the right.a. Which graph below best represents the position-versus-time graph for the motionalong the track? Explain your reasoning. (Hint: Think of motion as onedimensional.)b. Which graph below best represents the instantaneous velocity-versus-time graph?Explain your reasoning.c. Which graph below best represents the instantaneous acceleration-versus-timegraph? Explain your reasoning.+++(a)t(b)t(c)t+++(d)t(e)t(f)t+++ttt-(g) - (h) - (i)

287PROBLEM #4: BOUNCINGYou have a summer job working for NASA to design a low cost landingsystem for a Mars mission. The payload will be surrounded by a bigpadded ball and dropped onto the surface. When it reaches the surface,it will simply bounce. The height and the distance of the bounce will getsmaller with each bounce so that it finally comes to rest on the surface.Your task is to determine how the ratio of the horizontal distancecovered by two successive bounces depends on the ratio of the heights ofeach bounce and the ratio of the horizontal components of the velocityof each bounce. After making the calculation you decide to check it inyour laboratory on Earth.?How does the ratio of the horizontal distance coveredby two successive bounces depend on the ratio of theheights of each bounce and the ratio of the horizontalcomponents of the velocity of each bounce?EQUIPMENTFor this problem, you will have a ball, a stopwatch, a meter stick, and acomputer with a video camera and an analysis application written inLabVIEW.PREDICTIONCalculate the ratio of the horizontal distance of two successive bounces ifyou know the ratio of the heights of the bounces and the ratio of thehorizontal components of the initial velocity of each bounce.Be sure to state your assumptions so your boss can tell if they arereasonable for the Mars mission.WARM UP QUESTIONSThe following questions should help you make the prediction.1. Draw a good picture of the situation including the velocity andacceleration vectors at all relevant times. Decide on a coordinatesystem to use. Make sure you define the positive and negativedirections. During what time interval does the ball have a motion

288that is easiest to calculate? Is the acceleration of the ball during thattime interval constant or is it changing? Why? What is therelationship between the acceleration of the ball before and after abounce? Are the time intervals for two successive bounces equal?Why or why not? Clearly label the horizontal distances and theheights for each of those time intervals. What reasonableassumptions will you probably need to make to solve this problem?How will you check these assumptions with your data?2. Write down all of the kinematic equations that apply to the timeintervals you selected under the assumptions you have made. Makesure you clearly distinguish the equations describing the horizontalmotion and those describing the vertical motion. These equationsare the tools you will use to solve the problem. Start the problem bycalculating the quantity you wish to find: the horizontal distancecovered between two successive bounces.3. Write down an equation that gives the horizontal distance that theball travels during the time between the first and second bounce. Forthat time interval, select an equation that gives the horizontal distancethat the ball travels between bounces as a function of the initialhorizontal velocity of the ball, its horizontal acceleration, and thetime between bounces.4. The only additional unknown in your equation is the time intervalbetween bounces. You can determine it from the vertical motion ofthe ball. Select an equation that gives the change of vertical positionof the ball between the first and second bounce as a function of theinitial vertical velocity of the ball, its vertical acceleration, and thetime between bounces.5. An additional unknown, the ball’s initial vertical velocity, has beenintroduced. Determine it from another equation giving the heightthat the ball bounces as a function of the initial vertical velocity of theball, its vertical acceleration, and the time from the bounce until itreaches that position. How is that time interval related to the timebetween bounces? Show that this is true.6. Combining the previous steps gives you an equation for thehorizontal distance of a bounce in terms of the ball’s horizontalvelocity, the height of the bounce, and the vertical acceleration of theball.7. Repeat the above process for the next bounce and take the ratio ofhorizontal distances to get your prediction.

289EXPLORATIONReview your lab journal from any previous problem requiring analyzing avideo of a falling ball.Position the camera and adjust it for optimal performance. Make sureeveryone in your group gets the chance to operate the camera and the computer.Practice bouncing the ball without spin until you can get at least two fullbounces to fill the video screen (or at least the undistorted part of thevideo screen). Three is better so you can check your results. It will takepractice and skill to get a good set of bounces. Everyone in the groupshould try to determine who is best at throwing the ball.Determine how much time it takes for the ball to have the number ofbounces you will video and estimate the number of video points you willget in that time. Is that enough points to make the measurement? Adjustthe camera position and screen size to give you enough data pointswithout dropping too many.Although the ball is the best item to use to calibrate the video, the imagequality due to its motion might make this difficult. Instead, you mightneed to place an object of known length in the plane of motion of theball, near the center of the ball’s trajectory, for calibration purposes.Where you place your reference object does make a difference to yourresults. Determine the best place to put the reference object forcalibration.Step through the video and determine which part of the ball is easiest toconsistently determine. When the ball moves rapidly you may see twoimages of the ball due to the interlaced scan of a TV camera. You shouldonly use one of the images.Write down your measurement plan.MEASUREMENTMake a video of the ball being tossed. Make sure you can see the ball inevery frame of the video.Digitize the position of the ball in enough frames of the video so thatyou have the sufficient data to accomplish your analysis. Make sure youset the scale for the axes of your graph so that you can see the data pointsas you take them. Use your measurements of total distance the ball

290travels and total time to determine the maximum and minimum value foreach axis before taking data.ANALYSISAnalyze the video to get the horizontal distance of two successivebounces, the height of the two bounces, and the horizontal componentsof the ball’s velocity for each bounce. You will probably want tocalibrate the video independently for each bounce so you can begin yourtime as close as possible to when the ball leaves the ground. The pointwhere the bounce occurs will usually not correspond to a video frametaken by the camera so some estimation is necessary to determine thisposition.Choose a function to represent the horizontal position-versus -timegraph and another for the vertical position graph for the first bounce.How can you estimate the values of the constants of the functions? Youcan waste a lot of time if you just try to guess the constants. Whatkinematic quantities do these constants represent? How can you tellwhere the bounce occurred from each graph? Determine the height andhorizontal distance for the first bounce.Choose a function to represent the velocity-versus-time graph for eachcomponent of the velocity for the first bounce. How can you calculatethe values of the constants of these functions from the functionsrepresenting the position-versus-time graphs? Check how well thisworks. You can also estimate the values of the constants from the graph.Just trying to guess the constants can waste a lot of your time. Whatkinematic quantities do these constants represent? How can you tellwhere the bounce occurred from each graph? Determine the initialhorizontal velocity of the ball for the first bounce. What is the horizontaland vertical acceleration of the ball between bounces? Does this agreewith your expectations?Repeat this analysis for the second bounce.What kinematic quantities are approximately the same for each bounce?How does that simplify your prediction equation?

291CONCLUSIONHow do your graphs compare to your predictions and method questions?What are the limitations on the accuracy of your measurements andanalysis?Will the ratio you calculated be the same on Mars as on Earth? Why?What additional kinematic quantity, whose value you know, can bedetermined with the data you have taken to give you some indication ofthe precision of your measurement? How close is this quantity to itsknown value?

292PROBLEM #2:FORCES IN EQUILIBRIUMYou have a summer job with a research group studying the ecology of a rainforest in South America. To avoid walking on the delicate rain forest floor,the team members walk along a rope walkway that the local inhabitants havestrung from tree to tree through the forest canopy. Your supervisor isconcerned about the maximum amount of equipment each team membershould carry to safely walk from tree to tree. If the walkway sags too much,the team member could be in danger, not to mention possible damage to therain forest floor. You are assigned to set the load standards.Each end of the rope supporting the walkway goes over a branch and then isattached to a large weight hanging down. You need to determine how thesag of the walkway is related to the mass of a team member plus equipmentwhen they are at the center of the walkway between two trees. To checkyour calculation, you decide to model the situation using the equipmentshown below.?How does the vertical displacement of an objectsuspended on a string halfway between two branches,depend on the mass of that object?EQUIPMENTThe system consists of a central object, B, suspended halfway betweentwo pulleys by a string. The whole system is in equilibrium. The picturebelow is similar to the situation with which you will work. The objects Aand C, which have the same mass, allow you to determine the forceexerted on the central object by the string.You do need to make someLassumptions about whatyou can neglect. For thisPinvestigation, you will alsoneed a meter stick, twopulley clamps, three masshangers and a mass set tovary the mass of objects.

293PREDICTIONCalculate the change in the vertical displacement of the central object Bas you increase its mass. You should obtain an equation that predictshow the vertical displacement of central object B depends on its mass,the mass of objects A and C, and the horizontal distance between the twopulleys.Use your equation to make a graph of the vertical displacement of objectB as a function of its mass.WARM UP QUESTIONSTo solve this problem it is useful to have an organized problem-solvingstrategy such as the one outlined in the following questions. You shoulduse a technique similar to that used in Problem 1 (where a more detailedset of Warm Up Questions is given) to solve this problem. You mightalso find the Problem Solving section 4-6 of your textbook is useful.1. Draw a sketch similar to the one in the Equipment section. Drawvectors that represent the forces on objects A, B, C, and point P.Use trigonometry to show how the vertical displacement of object Bis related to the horizontal distance between the two pulleys and theangle that the string between the two pulleys sags below thehorizontal.2. The "known" (measurable) quantities in this problem are the massesof the objects and the distance between the pulleys; the unknownquantity is the vertical displacement of object B.3. Use Newton's laws to solve this problem. Write down theacceleration for each object. Draw separate force diagrams forobjects A, B, C and for point P (if you need help, see your text).What assumptions are you making?Which angles between your force vectors and your horizontalcoordinate axis are the same as the angle between the strings and thehorizontal?4. For each force diagram, write Newton's second law along eachcoordinate axis.5. Solve your equations to predict how the vertical displacement ofobject B depends on its mass, the mass of objects A and C, and thehorizontal distance between the two pulleys. Use this resultingequation to make a graph of how the vertical displacement changes asa function of the mass of object B.

2946. From your resulting equation, analyze what is the limit of mass ofobject B corresponding to the fixed mass of object A and C. Whatwill happen if the mass of object B is larger than twice the mass of A?EXPLORATIONStart with just the string suspended between the pulleys (no centralobject), so that the string looks horizontal. Attach a central object andobserve how the string sags. Decide on the origin from which you willmeasure the vertical position of the object.Try changing the mass of objects A and C (keep them equal for themeasurements but you will want to explore the case where they are notequal).Do the pulleys behave in a frictionless way for the entire range of weightsyou will use? How can you determine if the assumption of frictionlesspulleys is a good one?Add mass to the central object to decide what increments of mass willgive a good range of values for the measurement. Decide howmeasurements you will need to make.MEASUREMENTMeasure the vertical position of the central object as you increase itsmass. Make a table and record your measurements.ANALYSISMake a graph of the vertical displacement of the central object as afunction of its mass based on your measurements. On the same graph,plot your predicted equation.Where do the two curves match? Where do the two curves start todiverge from one another? What does this tell you about the system?What are the limitations on the accuracy of your measurements andanalysis?

295CONCLUSIONWhat will you report to your supervisor? How does the verticaldisplacement of an object suspended on a string between two pulleysdepend on the mass of that object? Did your measurements of thevertical displacement of object B agree with your initial predictions? Ifnot, why? State your result in the most general terms supported by youranalysis.What information would you need to apply your calculation to thewalkway through the rain forest?Estimate reasonable values for the information you need, and solve theproblem for the walkway over the rain forest.

296PROBLEM #5:CONSERVATION OF ANGULAR MOMENTUMWhile driving around the city, your car is constantly shifting gears. Youwonder how the gear shifting process works. Your friend tells you thatthere are gears in the transmission of your car that are rotating about thesame axis. When the car shifts, one of these gear assemblies is broughtinto connection with another one that drives the car’s wheels. Thinkingabout a car starting up, you decide to calculate how the angular speed ofa spinning object changes when it is brought into contact with anotherobject at rest. To keep your calculation simple, you decide to use a diskfor the initially spinning object and a ring for the object initially at rest.Both objects will be able to rotate freely about the same axis that iscentered on both objects. To test your calculation you decide to build alaboratory model of the situation.?What is the final angular velocity of a disk with an initialangular speed after being connected with a ring initiallyat rest?EQUIPMENTYou will use the same basic equipment in the previous problems.PREDICTIONCalculate, in terms of the initial angular velocity of the disk before thering is dropped onto the spinning disk and the characteristics of the

297system, the angular velocity of the disk after the ring is dropped onto theinitially spinning disk.WARM UP QUESTIONSTo complete your prediction, it is useful to use a problem solving strategy such as theone outlined below:1. Make two side view drawings of the situation (similar to the diagramin the Equipment section), one just as the ring is released, and oneafter the ring lands on the disk. Label all relevant kinematicquantities and write down the relationships that exist between them.Label all relevant forces.2. Determine the basic principles of physics that you will use and howyou will use them. Determine your system. Are any objects fromoutside your system interacting with your system? Write down yourassumptions and check to see if they are reasonable.3. Use conservation of angular momentum to determine the finalangular speed of the rotating objects. Why not use conservation ofenergy or conservation of momentum? Define your system and writethe conservation of angular momentum equation for this situation:What is the angular momentum of the system as the ring is released? Whatis its angular momentum after the ring connects with the disk? Is any significantangular momentum transferred to or from the system? If so, can you determine it orredefine your system so that there is no transfer?4. Identify the target quantity you wish to determine. Use the equationscollected in steps 1 and 3 to plan a solution for the target. If thereare more unknowns than equations, reexamine the previous steps tosee if there is additional information about the situation that can beexpressed in an addition equation. If not, see if one of the unknownswill cancel out.EXPLORATIONPractice dropping the ring into the groove on the disk as gently aspossible to ensure the best data. What happens if the ring is droppedoff-center? What happens if the disk does not fall smoothly into thegroove? Explain your answers.Decide what measurements you need to make to check your prediction.If any major assumptions are used in your calculations, decide on theadditional measurements that you need to make to justify them.Outline your measurement plan.

298Make some rough measurements to be sure your plan will work.MEASUREMENTSFollow your measurement plan. What are the uncertainties in yourmeasurements?ANALYSISDetermine the initial and final angular velocity of the disk from the datayou collected. Be sure to use an analysis technique that makes the mostefficient use of your data and your time. Using your prediction equationand your measured initial angular velocity, calculate the final angularvelocity of the disk. If your calculation incorporates any assumptions,make sure you justify these assumptions based on data that you haveanalyzed.CONCLUSIONDid your measurement of the final angular velocity agree with yourcalculated value by prediction? Why or why not? What are thelimitations on the accuracy of your measurements and analysis?Could you have used conservation of energy to solve this problem? Whyor why not? Use your data to check your answer.

299PROBLEM #3: MEASURING THE MAGNETICFIELD OF PERMANENT MAGENTSYou are now ready to measure the magnetic field near high-voltagepower lines. Before making this measurement, you decide to practice byusing your Hall probe on a bar magnet. Since you already know the mapof the magnetic field of a bar magnet, you decide to use the Hall probe todetermine how the magnitude of the magnetic field varies as you moveaway from the magnet along each of its axes. While thinking about thismeasurement you wonder if a bar magnet’s magnetic field might be theresult of the sum of the magnetic field of each pole. Although, to date,no isolated magnetic monopoles have ever been discovered, you wonderif you can model the situation as two magnetic monopoles, one at eachend of the magnet. Is it possible that the magnetic field from a singlemagnetic pole, a monopole, if they exist, has the same behavior as theelectric field from a point charge? You decide to check it out.?How does the magnitude of the magnetic field from abar magnet along each of its axes depend on the distancefrom the magnet? Is that behavior consistent with thedependence of the magnetic field on the distance awayfrom a single pole, being the same as the electric fieldfrom a point charge?EQUIPMENTYou will have a bar magnet, a meter stick, a Hall probe (see AppendixD), and a computer data acquisition system (see Appendix E). You willalso have a Taconite plate and a compass.PREDICTIONCalculate the magnetic field strength as a function of distance along eachaxis of a bar magnet. Make a graph of this function for each axis. Howdo you expect these graphs to compare to similar graphs of the electricfield along each axis of an electric dipole?

300WARM UP QUESTIONS1. Draw a bar magnet as a magnetic dipole consisting of two magneticmonopoles of equal strength but opposite sign, separated by somedistance. Label each monopole with its strength and sign. Label thedistance. Choose a convenient coordinate system.2. Select a point along one of the coordinate axes, outside the magnet,at which you will calculate the magnetic field. Determine the positionof that point with respect to your coordinate system. Determine thedistance of your point to each pole of the magnet, in terms of theposition of your point with respect to your coordinate system.3. Assume that the magnetic field from a magnetic monopole isanalogous to the electric field from a point charge, i.e. the magneticgfield is proportional to 2rwhere g is a measure of the strength ofthe monopole. Determine the direction of the magnetic field fromeach pole at the point of interest.4. Calculate the magnitude of the each component of the magnetic fieldfrom each pole at the point of interest. Add the magnetic field(remember it is a vector) from each pole at that point to get themagnetic field at that point.5. Graph your resulting equation for the magnetic field strength alongthat axis as a function of position along the axis.6. Repeat the above steps for the other axis.EXPLORATIONUsing either a Taconite plate or a compass check that the magnetic fieldof the bar magnet appears to be a dipole.Start the Hall probe program and go through the Hall probe calibrationprocedure outlined in Appendix E. Be sure the switch on youramplification box agrees with the value on the computer.Take one of the bar magnets and use the probe to check out the variationof the magnetic field. Based on your previous determination of themagnetic field map, be sure to orient the Hall probe correctly. Where isthe field the strongest? The weakest? How far away from the barmagnet can you still measure the field with the probe?Write down a measurement plan.

301MEASUREMENTBased on your exploration, choose a scale for your graph of magneticfield strength against position that will include all of the points you willmeasure.Choose an axis of the bar magnet and take measurements of themagnetic field strength in a straight line along the axis of the magnet. Besure that the field is always perpendicular to the probe. Make sure apoint appears on the graph of magnetic field strength versus positionevery time you enter a data point. Use this graph to determine where youshould take your next data point to map out the function in the mostefficient manner.Repeat for each axis of the magnet.ANALYSISCompare the graph of your calculated magnetic field to that which youmeasured for each axis of symmetry of your bar magnet. Can you fityour prediction equation to your measurements by adjusting theconstants?CONCLUSIONAlong which axis of the bar magnet does the magnetic field fall offfaster? Did your measured graph agree with your predicted graph? Ifnot, why? State your results in the most general terms supported by youranalysis.How does the shape of the graph of magnetic field strength versusdistance compare to the shape of the graph of electric field strengthversus distance, for an electric dipole along each axis? Is it reasonable toassume that the functional form of the magnetic field of a monopole isthe same as that of an electric charge? Explain your reasoning.

302PROBLEM #6: MEASURING THE MAGNETICFIELD OF TWO PARALLEL COILSYou have a part time job working in a laboratory developing large liquidcrystal displays that could be used for very thin TV screens and computermonitors. The alignment of the liquid crystals is very sensitive tomagnetic fields. It is important that the material sample be in a fairlyuniform magnetic field for some crystal alignment tests. The laboratoryhas two nearly identical large coils of wire mounted so that the distancebetween them equals their radii. You have been asked to determine themagnetic field between them to see if it is suitable for the test.?For two large, parallel coils, what is the magnetic fieldon the axis, as a function of the distance from themiddle of the two coils?EQUIPMENTConnect two large coils to apower supply so that each coilhas the same current. Each coilhas 150 turns.You will have a digital Multimeter(DMM), a compass, a meter stick,and a Hall probe. A computer isused for data acquisition.IRIRxPREDICTIONCalculate the magnitude of the magnetic field for two coils as a functionof the position along their central axis, for the special case where thedistance between the coils is the same as the radius of the coils. Use thisexpression to graph the magnetic field strength versus position along theaxis.

303WARM UP QUESTIONS1. Draw a picture of the situation showing the direction of the currentthrough each coil of wire. Establish a single convenient coordinatesystem for both coils.Label all of the relevant quantities.2. Select a point along the axis of the two coils at which you willdetermine an equation for the magnetic field. In the previousproblem, you calculated the magnetic field caused by one coil as afunction of the position along its axis. To solve this problem, addthe magnetic field from each coil at the selected point along the axis.Remember to pay attention to the geometry of your drawing. Theorigin of your coordinate system for this problem cannot be at thecenter of both coils at once. Also remember that the magnetic fieldis a vector.3. Use your equation to graph the magnetic field strength as a functionof position from the common origin along the central axis of thecoils. Describe the qualitative behavior of the magnetic fieldbetween the two coils. What about the region outside the coils?EXPLORATIONWARNING: You will be working with a power supply that cangenerate large electric voltages. Improper use can cause painful burns.To avoid danger, the power should be turned OFF and youshould WAIT at least one minute before any wires aredisconnected from or connected to the power supply. Nevergrasp a wire by its metal ends.Connect the large coils to the power supply with the current flowing in theopposite direction in both coils, using the adjustable voltage. Using yourcompass, explore the magnetic field produced. Be sure to look bothbetween the coils and outside the coils.Now connect the large coils to the power supply with the current flowing inthe same direction in both coils, using the adjustable voltage. Using yourcompass, explore the magnetic field produced. Be sure to look bothbetween the coils and outside the coils.

304Based on your observations, should the currents be in the same directionor in opposite directions to give the most uniform magnetic fieldbetween the coils?Connect the Hall probe according to the directions in Appendices D andE. For the current configuration that gives the most uniform magneticfield between the coils, explore the strength of the magnetic field alongthe axis between the coils. Follow the axis through the coils. Is the fieldstronger between or outside the coils? Where is the field strongestbetween the coils? The weakest?See how the field varies when you are between the two coils but moveoff the axis. How far from the axis of the coils can you measure thefield? Is it the same on both sides of the coils? Decide whether youshould set the amplifier to high or low sensitivity.When using the Hall probe program, consider where you want your zeroposition to be, so that you can compare to your prediction.Write down a measurement plan.MEASUREMENTBased on your exploration, choose a scale for your graph of magneticfield strength against position that will include all of the points you willmeasure.Use the Hall probe to measure the magnitude of the magnetic field alongthe axis of the coils of wire. Be sure to measure the field on both sidesof the coils.What are the units of your measured magnetic fields? How do thesecompare to the units of your prediction equations?Use the DMM to measure the current in the two coils.As a check, repeat these measurements with the other currentconfiguration.ANALYSISGraph the measured magnetic field of the coil along its axis as a functionof position and compare to your prediction.

305CONCLUSIONFor two large, parallel coils, how does the magnetic field on the axis varyas a function of distance along the axis? Did your measured values agreewith your predicted values? If not, why not? What are the limitations onthe accuracy of your measurements and analysis?Does this two-coil configuration satisfy the requirement of giving a fairlyuniform field? Over how large a region is the field constant to within20%? This very useful geometric configuration of two coils (distancebetween them equals their radius) is called a Helmholtz coil